-
Notifications
You must be signed in to change notification settings - Fork 2
/
Copy paths12-02-arithmetic-sequences-and-serie.html
1100 lines (1083 loc) · 149 KB
/
s12-02-arithmetic-sequences-and-serie.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<link href="shared/bookhub.css" rel="stylesheet" type="text/css">
<title>Arithmetic Sequences and Series</title>
</head>
<body>
<div id=navbar-top class="navbar">
<div class="navbar-part left">
<a href="s12-01-introduction-to-sequences-and-.html"><img src="shared/images/batch-left.png"></a> <a href="s12-01-introduction-to-sequences-and-.html">Previous Section</a>
</div>
<div class="navbar-part middle">
<a href="index.html"><img src="shared/images/batch-up.png"></a> <a href="index.html">Table of Contents</a>
</div>
<div class="navbar-part right">
<a href="s12-03-geometric-sequences-and-series.html">Next Section</a> <a href="s12-03-geometric-sequences-and-series.html"><img src="shared/images/batch-right.png"></a>
</div>
</div>
<div id="book-content">
<div class="section" id="fwk-redden-ch09_s02" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">9.2</span> Arithmetic Sequences and Series</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch09_s02_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch09_s02_o01" numeration="arabic">
<li>Identify the common difference of an arithmetic sequence.</li>
<li>Find a formula for the general term of an arithmetic sequence.</li>
<li>Calculate the <em class="emphasis">n</em>th partial sum of an arithmetic sequence.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch09_s02_s01" version="5.0" lang="en">
<h2 class="title editable block">Arithmetic Sequences</h2>
<p class="para editable block" id="fwk-redden-ch09_s02_s01_p01">An <span class="margin_term"><a class="glossterm">arithmetic sequence</a><span class="glossdef">A sequence of numbers where each successive number is the sum of the previous number and some constant <em class="emphasis">d</em>.</span></span>, or <span class="margin_term"><a class="glossterm">arithmetic progression</a><span class="glossdef">Used when referring to an arithmetic sequence.</span></span>, is a sequence of numbers where each successive number is the sum of the previous number and some constant <em class="emphasis">d</em>.</p>
<p class="para block" id="fwk-redden-ch09_s02_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0279" display="block"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>d</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>A</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>h</mi><mi>m</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>S</mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi></mstyle></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch09_s02_s01_p03">And because <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0280" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>d</mi></mrow></math></span>, the constant <em class="emphasis">d</em> is called the <span class="margin_term"><a class="glossterm">common difference</a><span class="glossdef">The constant <em class="emphasis">d</em> that is obtained from subtracting any two successive terms of an arithmetic sequence; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0281" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>d</mi></mrow><mo>.</mo></math></span></span></span>. For example, the sequence of positive odd integers is an arithmetic sequence,</p>
<p class="para block" id="fwk-redden-ch09_s02_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0282" display="block"><mrow><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>9</mn><mo>,</mo><mo>…</mo></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch09_s02_s01_p05">Here <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0283" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></math></span> and the difference between any two successive terms is 2. We can construct the general term <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0284" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mn>2</mn></mrow></math></span> where,</p>
<p class="para block" id="fwk-redden-ch09_s02_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0285" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>2</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mo>=</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>3</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>5</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>4</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>+</mo><mn>2</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>7</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>5</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>4</mn></msub><mo>+</mo><mn>2</mn><mo>=</mo><mn>7</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>9</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⋮</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch09_s02_s01_p07">In general, given the first term <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0286" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></math></span> of an arithmetic sequence and its common difference <em class="emphasis">d</em>, we can write the following:</p>
<p class="para block" id="fwk-redden-ch09_s02_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0287" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>2</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mi>d</mi></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>3</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><mi>d</mi><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mi>d</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>d</mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mi>d</mi></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>4</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>+</mo><mi>d</mi><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mi>d</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>d</mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><mi>d</mi></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>5</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>4</mn></msub><mo>+</mo><mi>d</mi><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><mi>d</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>d</mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>4</mn><mi>d</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⋮</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch09_s02_s01_p09">From this we see that any arithmetic sequence can be written in terms of its first element, common difference, and index as follows:</p>
<p class="para block" id="fwk-redden-ch09_s02_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0288" display="block"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>A</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>h</mi><mi>m</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>S</mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi></mstyle></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch09_s02_s01_p11">In fact, any general term that is linear in <em class="emphasis">n</em> defines an arithmetic sequence.</p>
<div class="callout block" id="fwk-redden-ch09_s02_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch09_s02_s01_p12">Find an equation for the general term of the given arithmetic sequence and use it to calculate its 100<sup class="superscript">th</sup> term: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0289" display="inline"><mrow><mn>7</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>16</mn><mo>,</mo><mn>19</mn><mo>,</mo><mo>…</mo></mrow></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p13">Begin by finding the common difference,</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p14"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0290" display="block"><mrow><mi>d</mi><mo>=</mo><mn>10</mn><mo>−</mo><mn>7</mn><mo>=</mo><mn>3</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p15">Note that the difference between any two successive terms is 3. The sequence is indeed an arithmetic progression where <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0291" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>7</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0292" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p16"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0293" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>7</mn><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>7</mn><mo>+</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>3</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p17">Therefore, we can write the general term <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0294" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow><mo>.</mo></math></span> Take a minute to verify that this equation describes the given sequence. Use this equation to find the 100<sup class="superscript">th</sup> term:</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p18"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0295" display="block"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>100</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>4</mn><mo>=</mo><mn>304</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p19">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0296" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0297" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mn>304</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch09_s02_s01_p20">The common difference of an arithmetic sequence may be negative.</p>
<div class="callout block" id="fwk-redden-ch09_s02_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch09_s02_s01_p21">Find an equation for the general term of the given arithmetic sequence and use it to calculate its 75<sup class="superscript">th</sup> term: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0298" display="inline"><mrow><mn>6</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mo>…</mo></mrow></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p22">Begin by finding the common difference,</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p23"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0299" display="block"><mrow><mi>d</mi><mo>=</mo><mn>4</mn><mo>−</mo><mn>6</mn><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p24">Next find the formula for the general term, here <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0300" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>6</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0301" display="inline"><mrow><mi>d</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p25"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0302" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mo>−</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>8</mn><mo>−</mo><mn>2</mn><mi>n</mi></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p26">Therefore, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0303" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>8</mn><mo>−</mo><mn>2</mn><mi>n</mi></mrow></math></span> and the 75<sup class="superscript">th</sup> term can be calculated as follows:</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p27"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0304" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mrow><mn>75</mn></mrow></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>8</mn><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>75</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>8</mn><mo>−</mo><mn>150</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>142</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p28">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0305" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>8</mn><mo>−</mo><mn>2</mn><mi>n</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0306" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>142</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch09_s02_s01_p29">The terms between given terms of an arithmetic sequence are called <span class="margin_term"><a class="glossterm">arithmetic means</a><span class="glossdef">The terms between given terms of an arithmetic sequence.</span></span>.</p>
<div class="callout block" id="fwk-redden-ch09_s02_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch09_s02_s01_p30">Find all terms in between <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0307" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>8</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0308" display="inline"><mrow><msub><mi>a</mi><mn>7</mn></msub><mo>=</mo><mn>10</mn></mrow></math></span> of an arithmetic sequence. In other words, find all arithmetic means between the 1<sup class="superscript">st</sup> and 7<sup class="superscript">th</sup> terms.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p31">Begin by finding the common difference <em class="emphasis">d</em>. In this case, we are given the first and seventh term:</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p32"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0309" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></mtd><mtd><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>U</mi><mi>s</mi><mi>e</mi><mtext> </mtext><mi>n</mi><mo>=</mo><mn>7</mn><mo>.</mo></mstyle></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>7</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mn>7</mn><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>7</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>6</mn><mi>d</mi></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p33">Substitute <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0310" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>8</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0311" display="inline"><mrow><msub><mi>a</mi><mn>7</mn></msub><mo>=</mo><mn>10</mn></mrow></math></span> into the above equation and then solve for the common difference <em class="emphasis">d</em>.</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p34"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0312" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mn>10</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn><mo>+</mo><mn>6</mn><mi>d</mi></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>18</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mi>d</mi></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>d</mi></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p35">Next, use the first term <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0313" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>8</mn></mrow></math></span> and the common difference <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0314" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow></math></span> to find an equation for the <em class="emphasis">n</em>th term of the sequence.</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p36"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0315" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn><mo>+</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>11</mn><mo>+</mo><mn>3</mn><mi>n</mi></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p37">With <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0316" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>11</mn></mrow></math></span>, where <em class="emphasis">n</em> is a positive integer, find the missing terms.</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p38"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0317" display="block"><mtable columnspacing="0.1em"><mtr><mtd><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mn>11</mn><mo>=</mo><mn>3</mn><mo>−</mo><mn>11</mn><mo>=</mo><mo>−</mo><mn>8</mn><mspace width="10.1em"></mspace></mtd></mtr><mtr><mtd><mrow><mtable columnspacing="0.1em"><mtr><mtd><mspace width="0.5em"></mspace><msub><mi>a</mi><mn>2</mn></msub><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mo>−</mo><mn>11</mn><mo>=</mo><mn>6</mn><mo>−</mo><mn>11</mn><mo>=</mo><mo>−</mo><mn>5</mn></mtd></mtr><mtr><mtd><mspace width="0.6em"></mspace><msub><mi>a</mi><mn>3</mn></msub><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mo>−</mo><mn>11</mn><mo>=</mo><mn>9</mn><mo>−</mo><mn>11</mn><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>4</mn></msub><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mo>−</mo><mn>11</mn><mo>=</mo><mn>12</mn><mo>−</mo><mn>11</mn><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>5</mn></msub><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow><mo>−</mo><mn>11</mn><mo>=</mo><mn>15</mn><mo>−</mo><mn>11</mn><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mspace width="0.75em"></mspace><msub><mi>a</mi><mn>6</mn></msub><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow><mo>−</mo><mn>11</mn><mo>=</mo><mn>18</mn><mo>−</mo><mn>11</mn><mo>=</mo><mn>7</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd></mtr></mtable><mo>}</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>a</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>h</mi><mi>m</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>m</mi><mi>e</mi><mi>a</mi><mi>n</mi><mi>s</mi></mstyle></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>7</mn></msub><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow><mo>−</mo><mn>11</mn><mo>=</mo><mn>21</mn><mo>−</mo><mn>11</mn><mo>=</mo><mn>10</mn><mspace width="10.1em"></mspace></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p39">Answer: −5, −2, 1, 4, 7</p>
</div>
<p class="para editable block" id="fwk-redden-ch09_s02_s01_p40">In some cases, the first term of an arithmetic sequence may not be given.</p>
<div class="callout block" id="fwk-redden-ch09_s02_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch09_s02_s01_p41">Find the general term of an arithmetic sequence where <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0318" display="inline"><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0319" display="inline"><mrow><msub><mi>a</mi><mrow><mn>10</mn></mrow></msub><mo>=</mo><mn>48</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p42">To determine a formula for the general term we need <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0320" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0321" display="inline"><mi>d</mi><mo>.</mo></math></span> A linear system with these as variables can be formed using the given information and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0322" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></math></span>:</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p43"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0323" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><msub><mi>a</mi><mn>3</mn></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mtd></mtr><mtr><mtd><msub><mi>a</mi><mrow><mn>10</mn></mrow></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mn>10</mn><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mtd></mtr></mtable></mrow></mrow></mtd><mtd><mrow><munder><mo>⇒</mo><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></munder></mrow></mtd><mtd><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd><mo>−</mo><mn>1</mn><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mi>d</mi></mtd></mtr><mtr><mtd><mn>48</mn><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>9</mn><mi>d</mi></mtd></mtr></mtable></mrow></mrow></mtd><mtd><mtable columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>U</mi><mi>s</mi><mi>e</mi><mtext> </mtext><msub><mi>a</mi><mn>3</mn></msub><mo>=</mo><mo>−</mo><mn>1</mn><mo>.</mo></mstyle></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>U</mi><mi>s</mi><mi>e</mi><mtext> </mtext><msub><mi>a</mi><mrow><mn>10</mn></mrow></msub><mo>=</mo><mn>48</mn><mo>.</mo></mstyle></mtd></mtr></mtable></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p44">Eliminate <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0324" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></math></span> by multiplying the first equation by −1 and add the result to the second equation.</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p45"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0325" display="block"><mtable columnalign="left"><mtr><mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mi>d</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>48</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>9</mn><mi>d</mi></mrow></mtd></mtr></mtable></mrow></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtable><mtr><mtd><mrow><mtext> </mtext><munder><mrow><mover><mo>⇒</mo><mrow><mstyle color="#007fbf"><mo>×</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mstyle><mtext> </mtext></mrow></mover></mrow><mrow><mtext> </mtext><mtext> </mtext></mrow></munder></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr></mtable><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><munder accentunder="true"><mrow><mtable><mtr><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>+</mo><mtext> </mtext><mtext> </mtext></mrow></mtd></mtr></mtable><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd><mrow><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo>−</mo><mn>2</mn><mi>d</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>48</mn></mrow></mtd><mtd><mrow><mo>=</mo><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mtext> </mtext><mtext> </mtext><mn>9</mn><mi>d</mi></mrow></mtd></mtr></mtable></mrow></mrow></mrow><mo stretchy="true">¯</mo></munder></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtable><mtr><mtd><mrow><mn>49</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><mn>7</mn><mi>d</mi></mrow></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mo>=</mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p46">Substitute <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0326" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>7</mn></mrow></math></span> into <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0327" display="inline"><mrow><mo>−</mo><mn>1</mn><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mi>d</mi></mrow></math></span> to find <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0328" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p47"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0329" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>14</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>−</mo><mn>15</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p48">Next, use the first term <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0330" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>15</mn></mrow></math></span> and the common difference <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0331" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>7</mn></mrow></math></span> to find a formula for the general term.</p>
<p class="para" id="fwk-redden-ch09_s02_s01_p49"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0332" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>15</mn><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>7</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>15</mn><mo>+</mo><mn>7</mn><mi>n</mi><mo>−</mo><mn>7</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>22</mn><mo>+</mo><mn>7</mn><mi>n</mi></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p50">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0333" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>7</mn><mi>n</mi><mo>−</mo><mn>22</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch09_s02_s01_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch09_s02_s01_p51"><strong class="emphasis bold">Try this!</strong> Find an equation for the general term of the given arithmetic sequence and use it to calculate its 100<sup class="superscript">th</sup> term: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0334" display="inline"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mn>2</mn><mo>,</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>,</mo><mn>3</mn><mo>,</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo>,</mo><mo>…</mo></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s01_p52">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0335" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0336" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mn>51</mn></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/_ovjvVKtKpQ" condition="http://img.youtube.com/vi/_ovjvVKtKpQ/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/_ovjvVKtKpQ" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch09_s02_s02" version="5.0" lang="en">
<h2 class="title editable block">Arithmetic Series</h2>
<p class="para block" id="fwk-redden-ch09_s02_s02_p01">An <span class="margin_term"><a class="glossterm">arithmetic series</a><span class="glossdef">The sum of the terms of an arithmetic sequence.</span></span> is the sum of the terms of an arithmetic sequence. For example, the sum of the first 5 terms of the sequence defined by <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0337" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> follows:</p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0338" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>S</mi><mn>5</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle displaystyle="true"><munderover><mo>Σ</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mn>5</mn></munderover><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>+</mo><mrow><mo>[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>+</mo><mrow><mo>[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>+</mo><mrow><mo>[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>4</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>+</mo><mrow><mo>[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>5</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>5</mn><mo>+</mo><mn>7</mn><mo>+</mo><mn>9</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>25</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p03">Adding 5 positive odd integers, as we have done above, is managable. However, consider adding the first 100 positive odd integers. This would be very tedious. Therefore, we next develop a formula that can be used to calculate the sum of the first <em class="emphasis">n</em> terms, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0339" display="inline"><mrow><msub><mi>S</mi><mi>n</mi></msub></mrow></math></span>, of any arithmetic sequence. In general,</p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0340" display="block"><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mi>d</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mi>d</mi></mrow><mo>)</mo></mrow><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch09_s02_s02_p05">Writing this series in reverse we have,</p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0341" display="block"><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>−</mo><mi>d</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>−</mo><mn>2</mn><mi>d</mi></mrow><mo>)</mo></mrow><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p07">And adding these two equations together, the terms involving <em class="emphasis">d</em> add to zero and we obtain <em class="emphasis">n</em> factors of <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0342" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow></math></span>:</p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0343" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mn>2</mn><msub><mi>S</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo>+</mo><mo>…</mo><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>2</mn><msub><mi>S</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p09">Dividing both sides by 2 leads us the formula for the <span class="margin_term"><a class="glossterm"><em class="emphasis">n</em>th partial sum of an arithmetic sequence</a><span class="glossdef">The sum of the first <em class="emphasis">n</em> terms of an arithmetic sequence given by the formula: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0344" display="inline"><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></span></span>:</p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0345" display="block"><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p11">Use this formula to calculate the sum of the first 100 terms of the sequence defined by <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0346" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>.</mo></math></span> Here <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0347" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0348" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mn>199</mn></mrow><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch09_s02_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0349" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>S</mi><mrow><mn>100</mn></mrow></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>100</mn><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>100</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>199</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>10,000</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<div class="callout block" id="fwk-redden-ch09_s02_s02_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch09_s02_s02_p13">Find the sum of the first 50 terms of the given sequence: 4, 9, 14, 19, 24, …</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p14">Determine whether or not there is a common difference between the given terms.</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0350" display="block"><mrow><mi>d</mi><mo>=</mo><mn>9</mn><mo>−</mo><mn>4</mn><mo>=</mo><mn>5</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p16">Note that the difference between any two successive terms is 5. The sequence is indeed an arithmetic progression and we can write</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p17"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0351" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>5</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mo>+</mo><mn>5</mn><mi>n</mi><mo>−</mo><mn>5</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p18">Therefore, the general term is <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0352" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>5</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>.</mo></math></span> To calculate the 50<sup class="superscript">th</sup> partial sum of this sequence we need the 1<sup class="superscript">st</sup> and the 50<sup class="superscript">th</sup> terms:</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p19"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0353" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mrow><mn>50</mn></mrow></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mn>50</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn><mo>=</mo><mn>249</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p20">Next use the formula to determine the 50<sup class="superscript">th</sup> partial sum of the given arithmetic sequence.</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p21"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0354" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>S</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>n</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>S</mi><mrow><mn>50</mn></mrow></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>50.</mn><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mrow><mn>50</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>50</mn><mo stretchy="false">(</mo><mn>4</mn><mo>+</mo><mn>249</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>25</mn><mo stretchy="false">(</mo><mn>253</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6,325</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p22">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0355" display="inline"><mrow><msub><mi>S</mi><mrow><mn>50</mn></mrow></msub><mo>=</mo><mn>6,325</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch09_s02_s02_n02">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch09_s02_s02_p23">Evaluate: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0356" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo>Σ</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>35</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mn>10</mn><mo>−</mo><mn>4</mn><mi>n</mi></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p24">In this case, we are asked to find the sum of the first 35 terms of an arithmetic sequence with general term <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0357" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>10</mn><mo>−</mo><mn>4</mn><mi>n</mi></mrow><mo>.</mo></math></span> Use this to determine the 1<sup class="superscript">st</sup> and the 35<sup class="superscript">th</sup> term.</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p25"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0358" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>10</mn><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>6</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mrow><mn>35</mn></mrow></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>10</mn><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mrow><mn>35</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>130</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p26">Next use the formula to determine the 35<sup class="superscript">th</sup> partial sum.</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p27"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0359" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>S</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>S</mi><mrow><mn>35</mn></mrow></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>35</mn><mo>⋅</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mrow><mn>35</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>35</mn><mrow><mo>[</mo><mrow><mn>6</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>130</mn></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>35</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>124</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2,170</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p28">Answer: −2,170</p>
</div>
<div class="callout block" id="fwk-redden-ch09_s02_s02_n03">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch09_s02_s02_p29">The first row of seating in an outdoor amphitheater contains 26 seats, the second row contains 28 seats, the third row contains 30 seats, and so on. If there are 18 rows, what is the total seating capacity of the theater?</p>
<div class="figure medium" id="fwk-redden-ch09_s02_s02_f01">
<p class="title"><span class="title-prefix">Figure 9.2</span> </p>
<img src="section_12/8a632bb6439e44faef3263c3ac8c12eb.png">
<p class="para">Roman Theater (Wikipedia)</p>
</div>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p30">Begin by finding a formula that gives the number of seats in any row. Here the number of seats in each row forms a sequence:</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p31"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0360" display="block"><mrow><mn>26</mn><mo>,</mo><mn>28</mn><mo>,</mo><mn>30</mn><mo>,</mo><mo>…</mo></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p32">Note that the difference between any two successive terms is 2. The sequence is an arithmetic progression where <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0361" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>26</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0362" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p33"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0363" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>26</mn><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>26</mn><mo>+</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>24</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p34">Therefore, the number of seats in each row is given by <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0364" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>24</mn></mrow><mo>.</mo></math></span> To calculate the total seating capacity of the 18 rows we need to calculate the 18<sup class="superscript">th</sup> partial sum. To do this we need the 1<sup class="superscript">st</sup> and the 18<sup class="superscript">th</sup> terms:</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p35"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0365" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>26</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>a</mi><mrow><mn>18</mn></mrow></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>18</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>24</mn><mo>=</mo><mn>60</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p36">Use this to calculate the 18<sup class="superscript">th</sup> partial sum as follows:</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p37"><span class="informalequation"><math xml:id="fwk-redden-ch09_m0366" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="right"><mtd columnalign="left"><mrow><msub><mi>S</mi><mi>n</mi></msub></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msub><mi>S</mi><mrow><mn>18</mn></mrow></msub></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>18</mn><mo>⋅</mo><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mrow><mn>18</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>18</mn><mrow><mo>(</mo><mrow><mn>26</mn><mo>+</mo><mn>60</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>9</mn><mrow><mo>(</mo><mrow><mn>86</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>774</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch09_s02_s02_p38">Answer: There are 774 seats total.</p>
</div>
<div class="callout block" id="fwk-redden-ch09_s02_s02_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch09_s02_s02_p39"><strong class="emphasis bold">Try this!</strong> Find the sum of the first 60 terms of the given sequence: 5, 0, −5, −10, −15, …</p>
<p class="para" id="fwk-redden-ch09_s02_s02_p40">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0367" display="inline"><mrow><msub><mi>S</mi><mrow><mn>60</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>8,550</mn></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/baYq2a_kBKo" condition="http://img.youtube.com/vi/baYq2a_kBKo/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/baYq2a_kBKo" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch09_s02_s02_n04">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch09_s02_s02_l01" mark="bullet">
<li>An arithmetic sequence is a sequence where the difference <em class="emphasis">d</em> between successive terms is constant.</li>
<li>The general term of an arithmetic sequence can be written in terms of its first term <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0368" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></math></span>, common difference <em class="emphasis">d</em>, and index <em class="emphasis">n</em> as follows: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0369" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow><mo>.</mo></math></span>
</li>
<li>An arithmetic series is the sum of the terms of an arithmetic sequence.</li>
<li>The <em class="emphasis">n</em>th partial sum of an arithmetic sequence can be calculated using the first and last terms as follows: <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0370" display="inline"><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow><mo>.</mo></math></span>
</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch09_s02_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd01">
<h3 class="title">Part A: Arithmetic Sequences</h3>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch09_s02_qs01_p01"><strong class="emphasis bold">Write the first 5 terms of the arithmetic sequence given its first term and common difference. Find a formula for its general term.</strong></p>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0371" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>5</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0372" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0374" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>12</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0375" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0377" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>15</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0378" display="inline"><mrow><mi>d</mi><mo>=</mo><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0380" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>7</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0381" display="inline"><mrow><mi>d</mi><mo>=</mo><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0383" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0384" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0391" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0392" display="inline"><mrow><mi>d</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0397" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0398" display="inline"><mrow><mi>d</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0402" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0403" display="inline"><mrow><mi>d</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0409" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>1.8</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0410" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>0.6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0412" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>4.3</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0413" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>2.1</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd01_qd02" start="11">
<p class="para" id="fwk-redden-ch09_s02_qs01_p22"><strong class="emphasis bold">Given the arithmetic sequence, find a formula for the general term and use it to determine the 100<sup class="superscript">th</sup> term.</strong></p>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p23">3, 9, 15, 21, 27,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p25">3, 8, 13, 18, 23,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p27">−3, −7, −11, −15, −19,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p29">−6, −14, −22, −30, −38,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p31">−5, −10, −15, −20, −25,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p33">2, 4, 6, 8, 10,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p35"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0427" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0428" display="inline"><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0429" display="inline"><mrow><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0430" display="inline"><mrow><mfrac><mn>13</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0431" display="inline"><mrow><mfrac><mn>17</mn><mn>2</mn></mfrac></mrow></math></span>,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p37"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0434" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0435" display="inline"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0436" display="inline"><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0437" display="inline"><mrow><mfrac><mn>8</mn><mn>3</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0438" display="inline"><mrow><mfrac><mn>11</mn><mn>3</mn></mfrac></mrow></math></span>,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p39"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0441" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span>, 0, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0442" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0443" display="inline"><mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math></span>, −1,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p41"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0446" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0447" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0448" display="inline"><mrow><mo>−</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow></math></span>, −2, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0449" display="inline"><mrow><mo>−</mo><mfrac><mn>11</mn><mn>4</mn></mfrac></mrow></math></span>,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p43">0.8, 2, 3.2, 4.4, 5.6,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p45">4.4, 7.5, 10.6, 13.7, 16.8,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p47">Find the 50<sup class="superscript">th</sup> positive odd integer.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p49">Find the 50<sup class="superscript">th</sup> positive even integer.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p51">Find the 40<sup class="superscript">th</sup> term in the sequence that consists of every other positive odd integer: 1, 5, 9, 13,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p53">Find the 40<sup class="superscript">th</sup> term in the sequence that consists of every other positive even integer: 2, 6, 10, 14,…</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p55">What number is the term 355 in the arithmetic sequence −15, −5, 5, 15, 25,…?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p57">What number is the term −172 in the arithmetic sequence 4, −4, −12, −20, −28,…?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p59">Given the arithmetic sequence defined by the recurrence relation <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0456" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mn>5</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0457" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0458" display="inline"><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow></math></span>, find an equation that gives the general term in terms of <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0459" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></math></span> and the common difference <em class="emphasis">d</em>.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p61">Given the arithmetic sequence defined by the recurrence relation <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0461" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><mn>9</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0462" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0463" display="inline"><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow></math></span>, find an equation that gives the general term in terms of <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0464" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></math></span> and the common difference <em class="emphasis">d</em>.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd01_qd03" start="31">
<p class="para" id="fwk-redden-ch09_s02_qs01_p63"><strong class="emphasis bold">Given the terms of an arithmetic sequence, find a formula for the general term.</strong></p>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p64"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0466" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>6</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0467" display="inline"><mrow><msub><mi>a</mi><mn>7</mn></msub><mo>=</mo><mn>42</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p66"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0469" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0470" display="inline"><mrow><msub><mi>a</mi><mrow><mn>12</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p68"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0472" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>19</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0473" display="inline"><mrow><msub><mi>a</mi><mrow><mn>26</mn></mrow></msub><mo>=</mo><mn>56</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p70"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0475" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>9</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0476" display="inline"><mrow><msub><mi>a</mi><mrow><mn>31</mn></mrow></msub><mo>=</mo><mn>141</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p72"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0478" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0479" display="inline"><mrow><msub><mi>a</mi><mrow><mn>10</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>37</mn></mrow><mn>6</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p74"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0481" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0482" display="inline"><mrow><msub><mi>a</mi><mrow><mn>11</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>65</mn></mrow><mn>4</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p76"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0484" display="inline"><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>=</mo><mn>6</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0485" display="inline"><mrow><msub><mi>a</mi><mrow><mn>26</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>40</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p78"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0487" display="inline"><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>=</mo><mn>16</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0488" display="inline"><mrow><msub><mi>a</mi><mrow><mn>15</mn></mrow></msub><mo>=</mo><mn>76</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p80"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0490" display="inline"><mrow><msub><mi>a</mi><mn>4</mn></msub><mo>=</mo><mo>−</mo><mn>8</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0491" display="inline"><mrow><msub><mi>a</mi><mrow><mn>23</mn></mrow></msub><mo>=</mo><mn>30</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p82"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0493" display="inline"><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>=</mo><mo>−</mo><mn>7</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0494" display="inline"><mrow><msub><mi>a</mi><mrow><mn>37</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>135</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p84"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0496" display="inline"><mrow><msub><mi>a</mi><mn>4</mn></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mn>23</mn></mrow><mrow><mn>10</mn></mrow></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0497" display="inline"><mrow><msub><mi>a</mi><mrow><mn>21</mn></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mn>25</mn></mrow><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p86"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0499" display="inline"><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>8</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0500" display="inline"><mrow><msub><mi>a</mi><mrow><mn>12</mn></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mn>11</mn></mrow><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p88"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0502" display="inline"><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>=</mo><mn>13.2</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0503" display="inline"><mrow><msub><mi>a</mi><mrow><mn>26</mn></mrow></msub><mo>=</mo><mn>61.5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p90"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0505" display="inline"><mrow><msub><mi>a</mi><mn>4</mn></msub><mo>=</mo><mo>−</mo><mn>1.2</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0506" display="inline"><mrow><msub><mi>a</mi><mrow><mn>13</mn></mrow></msub><mo>=</mo><mn>12.3</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd01_qd04" start="45">
<p class="para" id="fwk-redden-ch09_s02_qs01_p92"><strong class="emphasis bold">Find all arithmetic means between the given terms.</strong></p>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0508" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0509" display="inline"><mrow><msub><mi>a</mi><mn>6</mn></msub><mo>=</mo><mn>17</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0510" display="inline"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>5</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0511" display="inline"><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>=</mo><mo>−</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0512" display="inline"><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>=</mo><mn>4</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0513" display="inline"><mrow><msub><mi>a</mi><mn>8</mn></msub><mo>=</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0517" display="inline"><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0518" display="inline"><mrow><msub><mi>a</mi><mn>9</mn></msub><mo>=</mo><mo>−</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0522" display="inline"><mrow><msub><mi>a</mi><mn>5</mn></msub><mo>=</mo><mn>15</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0523" display="inline"><mrow><msub><mi>a</mi><mn>7</mn></msub><mo>=</mo><mn>21</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0524" display="inline"><mrow><msub><mi>a</mi><mn>6</mn></msub><mo>=</mo><mn>4</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0525" display="inline"><mrow><msub><mi>a</mi><mrow><mn>11</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd02">
<h3 class="title">Part B: Arithmetic Series</h3>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd02_qd01" start="51">
<p class="para" id="fwk-redden-ch09_s02_qs01_p105"><strong class="emphasis bold">Calculate the indicated sum given the formula for the general term.</strong></p>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p106"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0526" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>5</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0527" display="inline"><mrow><msub><mi>S</mi><mrow><mn>100</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p108"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0528" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>5</mn><mi>n</mi><mo>−</mo><mn>11</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0529" display="inline"><mrow><msub><mi>S</mi><mrow><mn>100</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0530" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mi>n</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0531" display="inline"><mrow><msub><mi>S</mi><mrow><mn>70</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0532" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>n</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0533" display="inline"><mrow><msub><mi>S</mi><mrow><mn>120</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0534" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0535" display="inline"><mrow><msub><mi>S</mi><mrow><mn>20</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0536" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mi>n</mi><mo>−</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0537" display="inline"><mrow><msub><mi>S</mi><mrow><mn>150</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0538" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>45</mn><mo>−</mo><mn>5</mn><mi>n</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0539" display="inline"><mrow><msub><mi>S</mi><mrow><mn>65</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0540" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>48</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0541" display="inline"><mrow><msub><mi>S</mi><mrow><mn>95</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0542" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>4.4</mn><mo>−</mo><mn>1.6</mn><mi>n</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0543" display="inline"><mrow><msub><mi>S</mi><mrow><mn>75</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0544" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>6.5</mn><mi>n</mi><mo>−</mo><mn>3.3</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0545" display="inline"><mrow><msub><mi>S</mi><mrow><mn>67</mn></mrow></msub></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd02_qd02" start="61">
<p class="para" id="fwk-redden-ch09_s02_qs01_p126"><strong class="emphasis bold">Evaluate.</strong></p>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa61">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0546" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>160</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>n</mi></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa62">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0547" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>121</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mi>n</mi></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa63">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0548" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>250</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa64">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0549" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>120</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa65">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0550" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>70</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mn>19</mn><mo>−</mo><mn>8</mn><mi>n</mi></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa66">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0551" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>220</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mi>n</mi></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa67">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0552" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>60</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa68">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0553" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>51</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mn>8</mn></mfrac><mi>n</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa69">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0554" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>120</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mn>1.5</mn><mi>n</mi><mo>−</mo><mn>2.6</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa70">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch09_m0555" display="block"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>175</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>0.2</mn><mi>n</mi><mo>−</mo><mn>1.6</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p147">Find the sum of the first 200 positive integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p149">Find the sum of the first 400 positive integers.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd02_qd03" start="73">
<p class="para" id="fwk-redden-ch09_s02_qs01_p151"><strong class="emphasis bold">The general term for the sequence of positive odd integers is given by <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0556" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> and the general term for the sequence of positive even integers is given by <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0557" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi></mrow><mo>.</mo></math></span> Find the following.</strong></p>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p152">The sum of the first 50 positive odd integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p154">The sum of the first 200 positive odd integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p156">The sum of the first 50 positive even integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p158">The sum of the first 200 positive even integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p160">The sum of the first <em class="emphasis">k</em> positive odd integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p162">The sum of the first <em class="emphasis">k</em> positive even integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p164">The first row of seating in a small theater consists of 8 seats. Each row thereafter consists of 3 more seats than the previous row. If there are 12 rows, how many total seats are in the theater?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p166">The first row of seating in an outdoor amphitheater contains 42 seats, the second row contains 44 seats, the third row contains 46 seats, and so on. If there are 22 rows, what is the total seating capacity of the theater?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p168">If a triangular stack of bricks has 37 bricks on the bottom row, 34 bricks on the second row and so on with one brick on top. How many bricks are in the stack?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p170">Each successive row of a triangular stack of bricks has one less brick until there is only one brick on top. How many rows does the stack have if there are 210 total bricks?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p172">A 10-year salary contract offers $65,000 for the first year with a $3,200 increase each additional year. Determine the total salary obligation over the 10 year period.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p174">A clock tower strikes its bell the number of times indicated by the hour. At one o’clock it strikes once, at two o’clock it strikes twice and so on. How many times does the clock tower strike its bell in a day?</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch09_s02_qs01_qd03_qd01" start="85">
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p176">Is the Fibonacci sequence an arithmetic sequence? Explain.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p177">Use the formula for the <em class="emphasis">n</em>th partial sum of an arithmetic sequence <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0560" display="inline"><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mrow></math></span> and the formula for the general term <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0561" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow></math></span> to derive a new formula for the <em class="emphasis">n</em>th partial sum <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0562" display="inline"><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mrow><mo>[</mo><mrow><mn>2</mn><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>d</mi></mrow><mo>]</mo></mrow></mrow><mo>.</mo></math></span> Under what circumstances would this formula be useful? Explain using an example of your own making.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p178">Discuss methods for calculating sums where the index does not start at 1. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0563" display="inline"><mrow><mstyle displaystyle="true"><munderover><mo>Σ</mo><mrow><mi>n</mi><mo>=</mo><mn>15</mn></mrow><mrow><mn>35</mn></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>=</mo><mn>1,659</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch09_s02_qs01_p179">A famous story involves Carl Friedrich Gauss misbehaving at school. As punishment, his teacher assigned him the task of adding the first 100 integers. The legend is that young Gauss answered correctly within seconds. What is the answer and how do you think he was able to find the sum so quickly?</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch09_s02_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p03_ans">5, 8, 11, 14, 17; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0373" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p07_ans">15, 10, 5, 0, −5; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0379" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>20</mn><mo>−</mo><mn>5</mn><mi>n</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p11_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0385" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0386" display="inline"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0387" display="inline"><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0388" display="inline"><mrow><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0389" display="inline"><mrow><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0390" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mi>n</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p15_ans">1, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0399" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>, 0, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0400" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>, −1; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0401" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p19_ans">1.8, 2.4, 3, 3.6, 4.2; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0411" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>0.6</mn><mi>n</mi><mo>+</mo><mn>1.2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p24_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0415" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>6</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0416" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mn>597</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p28_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0419" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo>−</mo><mn>4</mn><mi>n</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0420" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>399</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p32_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0423" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mo>−</mo><mn>5</mn><mi>n</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0424" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>500</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p36_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0432" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0433" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>397</mn></mrow><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p40_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0444" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>n</mi></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0445" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mn>98</mn></mrow><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa21_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p44_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0452" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>1.2</mn><mi>n</mi><mo>−</mo><mn>0.4</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0453" display="inline"><mrow><msub><mi>a</mi><mrow><mn>100</mn></mrow></msub><mo>=</mo><mn>119.6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa23_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p48_ans">99</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa24_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa25_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p52_ans">157</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa26_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa27_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p56_ans">38</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa28_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa29_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p60_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0460" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>5</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa30_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa31_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p65_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0468" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>6</mn><mi>n</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa32_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa33_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p69_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0474" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>22</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa34_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa35_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p73_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0480" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>n</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa36_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa37_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p77_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0486" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>12</mn><mo>−</mo><mn>2</mn><mi>n</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa38_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa39_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p81_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0492" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>16</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa40_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa41_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p85_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0498" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><mo>−</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><mi>n</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa42_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa43_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p89_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0504" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>2.3</mn><mi>n</mi><mo>+</mo><mn>1.7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa44_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa45_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p94_ans">1, 5, 9, 13</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa46_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa47_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p98_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch09_m0514" display="inline"><mrow><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></math></span>, 5, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0515" display="inline"><mrow><mfrac><mn>11</mn><mn>2</mn></mfrac></mrow></math></span>, 6, <span class="inlineequation"><math xml:id="fwk-redden-ch09_m0516" display="inline"><mrow><mfrac><mn>13</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa48_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa49_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p102_ans">18</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa50_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="51">
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa51_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p107_ans">15,650</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa52_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa53_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p111_ans">−2,450</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa54_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa55_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p115_ans">90</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa56_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa57_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p119_ans">−7,800</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa58_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa59_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p123_ans">−4,230</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa60_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa61_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p128_ans">38,640</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa62_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa63_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p132_ans">124,750</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa64_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa65_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p136_ans">−18,550</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa66_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa67_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p140_ans">−765</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa68_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa69_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p144_ans">10,578</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa70_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa71_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p148_ans">20,100</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa72_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch09_s02_qs01_qa73_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch09_s02_qs01_p153_ans">2,500</p>
</div>
</li>