8-1: Arithmetic Sequences and Series Unit 8: Sequences/Series/Statistics English Casbarro.
CHAPTER 8 Sequences, Series, and...
Transcript of CHAPTER 8 Sequences, Series, and...
C H A P T E R 8Sequences, Series, and Probability
Section 8.1 Sequences and Series . . . . . . . . . . . . . . . . . . . . 690
Section 8.2 Arithmetic Sequences and Partial Sums . . . . . . . . . . 705
Section 8.3 Geometric Sequences and Series . . . . . . . . . . . . . . 713
Section 8.4 Mathematical Induction . . . . . . . . . . . . . . . . . . 724
Section 8.5 The Binomial Theorem . . . . . . . . . . . . . . . . . . 738
Section 8.6 Counting Principles . . . . . . . . . . . . . . . . . . . . . 748
Section 8.7 Probability . . . . . . . . . . . . . . . . . . . . . . . . . 753
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
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C H A P T E R 8Sequences, Series, and Probability
Section 8.1 Sequences and Series
690
■ Given the general th term in a sequence, you should be able to find, or list, some of the terms.
■ You should be able to find an expression for the th term of a sequence.
■ You should be able to use and evaluate factorials.
■ You should be able to use sigma notation for a sum.
n
n
1.
a5 � 2�5� � 5 � 15
a4 � 2�4� � 5 � 13
a3 � 2�3� � 5 � 11
a2 � 2�2� � 5 � 9
a1 � 2�1� � 5 � 7
an � 2n � 5 3.
a5 � 25 � 32
a4 � 24 � 16
a3 � 23 � 8
a2 � 22 � 4
a1 � 21 � 2
an � 2n
5.
a5 � ��12�5
� �132
a4 � ��12�4
�116
a3 � ��12�3
� �18
a2 � ��12�2
�14
a1 � ��12�1
� �12
an � ��12�n
2.
a5 � 4�5� � 7 � 13
a4 � 4�4� � 7 � 9
a3 � 4�3� � 7 � 5
a2 � 4�2� � 7 � 1
a1 � 4�1� � 7 � �3
an � 4n � 7
4.
a5 � � 12�5
�132
a4 � � 12�4
�116
a3 � � 12�3
�18
a2 � � 12�2
�14
a1 � � 12�1
�12
an � � 12�n
6.
a5 � ��2�5 � �32
a4 � ��2�4 � 16
a3 � ��2�3 � �8
a2 � ��2�2 � 4
a1 � ��2�1 � �2
an � ��2�n
Vocabulary Check
1. infinite sequence 2. terms 3. finite
4. recursively 5. factorial 6. summation notation
7. index, upper limit, lower limit 8. series 9. partial sumnth
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Section 8.1 Sequences and Series 691
10.
a5 �106
�53
a4 �85
a3 �64
�32
a2 �43
a1 �2�1�
1 � 1� 1
an �2n
n � 1 12.
a5 �1 � 12�5� � 0
a4 �1 � 12�4� �
14
a3 �1 � 12�3� � 0
a2 �1 � 12�2� �
12
a1 �1 � 1
2� 0
an �1 � ��1�n
2n11.
a5 � 0
a4 �2
4�
1
2
a3 � 0
a2 �2
2� 1
a1 � 0
an �1 � ��1�n
n
13.
a5 � 1 �125 �
3132
a4 � 1 �124 �
1516
a3 � 1 �123 �
78
a2 � 1 �122 � 1 �
14
�34
a1 � 1 �121 �
12
an � 1 �12n
15.
a5 �1
53�2
a4 �1
43�2�
1
8
a3 �1
33�2
a2 �1
23�2
a1 �1
1� 1
an �1
n3�214.
a5 �35
45�
243
1024
a4 �34
44�
81
256
a3 �33
43�
27
64
a2 �32
42�
9
16
a1 �31
41�
3
4
an �3n
4n
8.
a5 �5
5 � 1�
5
6
a4 �4
4 � 1�
4
5
a3 �3
3 � 1�
3
4
a2 �2
2 � 1�
2
3
a1 �1
1 � 1�
1
2
an �n
n � 17.
a5 �6
5
a4 �5
4
a3 �4
3
a2 �3
2
a1 �1 � 1
1� 2
an �n � 1
n9.
a5 �5
52 � 1�
526
a4 �4
42 � 1�
417
a3 �3
32 � 1�
310
a2 �2
22 � 1�
25
a1 �1
12 � 1�
12
an �n
n2 � 1
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692 Chapter 8 Sequences, Series, and Probability
20.
a5 � 5�5 � 1��5 � 2� � 60
a4 � 4�4 � 1��4 � 2� � 24
a3 � 3�3 � 1��3 � 2� � 6
a2 � 2�2 � 1��2 � 2� � 0
a1 � 1�1 � 1��1 � 2� � 0
an � n�n � 1��n � 2�
22. a16 � ��1�15 �16�15�� � �240
24.
a5 �52
2�5� � 1�
2511
an �n2
2n � 1
19.
a5 � �9��11� � 99
a4 � �7��9� � 63
a3 � �5��7� � 35
a2 � �3��5� � 15
a1 � �1��3� � 3
an � �2n � 1��2n � 1�
21. a25 � ��1�25�3�25� � 2� � �73 23. a10 �102
102 � 1�
100101
25. a6 �26
26 � 1�
6465
26. a7 �27�1
27 � 1�
28
27 � 1�
256129
27.
00 11
8
an �2
3 n 29.
−10
0 11
20
an � 16��0.5�n�1
31.
00 11
3
an �2n
n � 1
28.
−3
0 11
3
an � 2 �4
n
30.
00 11
10
an � 8�0.75�n�1 32.
00 11
5
an �3n2
n2 � 1
17.
a5 ��1
25
a4 �1
16
a3 ��1
9
a2 �1
4
a1 ��1
1� �1
an ���1�n
n216.
a5 �1�5
a4 �1�4
�12
a3 �1�3
a2 �1�2
a1 � 1
an �1�n
18.
a5 � ��1�55
5 � 1� �
5
6
a4 � ��1�44
4 � 1�
4
5
a3 � ��1�33
3 � 1� �
3
4
a2 � ��1�22
1 � 2�
2
3
a1 � ��1�11
1 � 1� �
1
2
an � ��1�n� n
n � 1�
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Section 8.1 Sequences and Series 693
36. an �4n2
�n � 2�
n 1 2 3 4 5 6 7 8 9 10
4 7.2 10.67 14.29 18 21.78 25.6 29.45 33.33 43an
35. an � 1 �n � 1
n
n 1 2 3 4 5 6 7 8 9 10
3 2.5 2.33 2.25 2.2 2.17 2.14 2.13 2.11 2.1an
37. an � ��1�n � 1
n 1 2 3 4 5 6 7 8 9 10
0 2 0 2 0 2 0 2 0 2an
38. an � ��1�n�1 � 1
n 1 2 3 4 5 6 7 8 9 10
2 0 2 0 2 0 2 0 2 0an
40.
Matches graph (b).
a1 � 4, a4 �8�4�
5�
32
5
an → 8 as n →�
an �8n
n � 139.
Matches graph (c).
a1 � 4, a10 �8
11
an → 0 as n →�
an �8
n � 141.
Matches graph (d).
a1 � 4, a10 0.008
an → 0 as n → �
an � 4�0.5�n�1
34. an � 2n�n � 1��n � 2�
n 1 2 3 4 5 6 7 8 9 10
12 48 120 240 420 672 1008 1440 1980 2640an
33. an � 2�3n � 1� � 5
n 1 2 3 4 5 6 7 8 9 10
9 15 21 27 33 39 45 51 57 63an
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694 Chapter 8 Sequences, Series, and Probability
48.
an �n � 1
2n � 1
2
1,
3
3,
4
5,
5
7,
6
9, . . . 50.
an ���1�n�12n�1
3n ���2�n�1
3n
1
3,
�2
9,
4
27,
�8
81, . . .49.
an ���1�n�1
2n
1
2,
�1
4,
1
8,
�1
16, . . .
51.
an � 1 �1
n
1 �1
1, 1 �
1
2, 1 �
1
3, 1 �
1
4, 1 �
1
5, . . . 52.
an � 1 �2n � 1
2n
1 �1
2, 1 �
3
4, 1 �
7
8, 1 �
15
16, 1 �
31
32, . . .
54.
an �2n�1
�n � 1�!
1, 2, 22
2,
23
6,
24
24,
25
120, . . .53.
an �1
n!
1, 1
2,
1
6,
1
24,
1
120, . . . 55.
an � 2 � ��1�n
1, 3, 1, 3, 1, 3, . . .
56.
an � ��1�n�1
1, �1, 1, �1, 1, �1, . . . 58.
a5 � a4 � 3 � 24 � 3 � 27
a4 � a3 � 3 � 21 � 3 � 24
a3 � a2 � 3 � 18 � 3 � 21
a2 � a1 � 3 � 15 � 3 � 18
a1 � 15
a1 � 15, ak�1 � ak � 3
60.
a5 �12a4 �
12�4� � 2
a4 �12a3 �
12�8� � 4
a3 �12a2 �
12�16� � 8
a2 �12a1 �
12�32� � 16
a1 � 32
a1 � 32, ak�1 �12ak
57.
a5 � a4 � 4 � 16 � 4 � 12
a4 � a3 � 4 � 20 � 4 � 16
a3 � a2 � 4 � 24 � 4 � 20
a2 � a1 � 4 � 28 � 4 � 24
a1 � 28
a1 � 28 and ak�1 � ak � 4
59.
a5 � 2�a4 � 1� � 2�10 � 1� � 18
a4 � 2�a3 � 1� � 2�6 � 1� � 10
a3 � 2�a2 � 1� � 2�4 � 1� � 6
a2 � 2�a1 � 1� � 2�3 � 1� � 4
a1 � 3
a1 � 3 and ak�1 � 2�ak � 1�
43.
an � 1 � �n � 1�3 � 3n � 2
1, 4, 7, 10, 13, . . .42.
Matches graph (a).
a1 � 4, a4 �44
4!�
256
24� 10
2
3
an → 0 as n →�
an �4n
n!44. 3, 7, 11, 15, 19, . . .
an � 4n � 1
45.
an � n2 � 1
0, 3, 8, 15, 24, . . . 47.
an �n � 1
n � 2
2
3,
3
4,
4
5,
5
6,
6
7, . . .46.
an �1
n2
1, 1
4,
1
9,
1
16,
1
25, . . .
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Section 8.1 Sequences and Series 695
63.
In general, an � 81�1
3�n�1
� 81�3��1
3�n
�243
3n.
a5 �1
3a4 �
1
3�3� � 1
a4 �1
3a3 �
1
3�9� � 3
a3 �1
3a2 �
1
3�27� � 9
a2 �1
3a1 �
1
3�81� � 27
a1 � 81
a1 � 81 and ak�1 �13
ak 64.
In general, an � 14��2�n�1.
a5 � ��2��a4� � ��2���112� � 224
a4 � ��2�a3 � ��2��56� � �112
a3 � ��2�a2 � ��2���28� � 56
a2 � ��2�a1 � ��2��14� � �28
a1 � 14
a1 � 14, ak�1 � ��2�ak
66.
a4 �1
5!�
1
120
a3 �1
4!�
1
24
a2 �1
3!�
1
6
a1 �1
2!�
1
2
a0 �11!
� 1
an �1
�n � 1�!65.
a4 �14!
�124
a3 �13!
�16
a2 �12
a1 �11!
� 1
a0 �10!
� 1
an �1n!
67.
a4 �4!
8 � 1�
249
�83
a3 �3!
6 � 1�
67
a2 �2!
4 � 1�
25
a1 �1!
2 � 1�
13
a0 �0!1
� 1
an �n!
2n � 1
61.
In general, an � 2n � 4.
a5 � a4 � 2 � 12 � 2 � 14
a4 � a3 � 2 � 10 � 2 � 12
a3 � a2 � 2 � 8 � 2 � 10
a2 � a1 � 2 � 6 � 2 � 8
a1 � 6
a1 � 6 and ak�1 � ak � 2 62.
In general, an � 30 � 5n .
a5 � a4 � 5 � 10 � 5 � 5
a4 � a3 � 5 � 15 � 5 � 10
a3 � a2 � 5 � 20 � 5 � 15
a2 � a1 � 5 � 25 � 5 � 20
a1 � 25
a1 � 25, ak�1 � ak � 5
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696 Chapter 8 Sequences, Series, and Probability
71.2!4!
�2!
4 � 3 � 2!�
112
73.
�12 � 11 � 10 � 9
4 � 3 � 2� 495
12!4!8!
�12 � 11 � 10 � 9 � 8!
4!8!
75.�n � 1�!
n!�
�n � 1�n!
n!� n � 1
77.
�1
2n�2n � 1�
�2n � 1�!�2n � 1�!
��2n � 1�!
�2n � 1��2n��2n � 1�!
76.�n � 2�!
n!�
�n � 2��n � 1�n!
n!� �n � 2��n � 1�
78.
� �2n � 2��2n � 1�
�2n � 2�!�2n�!
��2n � 2��2n � 1��2n�!
�2n�!
80. 6
i�1
�3i � 1� � �3 � 1 � 1� � �3 � 2 � 1� � �3 � 3 � 1� � �3 � 4 � 1� � �3 � 5 � 1� � �3 � 6 � 1� � 57
72.5!7!
�5!
7�6��5!� �142
74.
�10 � 9 � 8 � 7
4� 1260
10! 3!4! 6!
�10 � 9 � 8 � 7 � 6! � 3!
4 � 3! � 6!
79. 5
i�1
�2i � 1� � �2 � 1� � �4 � 1� � �6 � 1� � �8 � 1� � �10 � 1� � 35
81. 4
k�1
10 � 10 � 10 � 10 � 10 � 40
83. 4
i�0 i
2 � 02 � 12 � 22 � 32 � 42 � 30
85. 3
k�0
1
k2 � 1�
1
1�
1
1 � 1�
1
4 � 1�
1
9 � 1�
9
5
82. 5
k�1
6 � 6 � 6 � 6 � 6 � 6 � 30
84.
� 3�02 � 12 � 22 � 32 � 42 � 52� � 165
5
k�0
3i2 � 35
i�0
i2
86. 5
j�3
1
j�
1
3�
1
4�
1
5�
47
60
68.
a4 �16
5!�
16
120�
2
15
a3 �32
4!�
9
24�
3
8
a2 �22
3!�
2
3
a1 �1
2
a0 � 0
an �n2
�n � 1�!70.
a4 ��1
9!�
�1
362,880
a3 ��1
7!�
�1
5040
a2 ��1
5!�
�1
120
a1 ���1�3
3!�
�1
6
a0 ��11
1!� �1
an ���1�2n�1
�2n � 1�!69.
a4 ���1�8
8!�
140,320
a3 ���1�6
6!�
1720
a2 ���1�4
4!�
124
a1 ���1�2
2!�
12
a0 ���1�0
0!� 1
an ���1�2n
�2n�!
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Section 8.1 Sequences and Series 697
90. 4
j�0
��2�j � ��2�0 � ��2�1 � ��2�2 � ��2�3 � ��2�4 � 11
95.1
3�1��
1
3�2��
1
3�3�� . . . �
1
3�9��
9
i�1
1
3i 0.94299
96.5
1 � 1�
5
1 � 2�
5
1 � 3� . . . �
5
1 � 15�
15
i�1
5
1 � i 11.904
97. � 33�2�1
8� � 3� � �2�2
8� � 3� � �2�3
8� � 3� � . . . � �2�8
8� � 3� � 8
i�1 �2� i
8� � 3�
99. 3 � 9 � 27 � 81 � 243 � 729 � 6
i�1
��1�i�13i � �546
101.1
12�
1
22�
1
32�
1
42� . . . �
1
202�
20
i�1
��1�i�1
i2 0.82128
103.1
4�
3
8�
7
16�
15
32�
31
64�
5
i�1
2i � 1
2i�1�
129
64� 2.015625
98. �1 � �16�
2
� � �1 � �26�
2
� � . . . � �1 � �66�
2
� � 6
k�1�1 � �k
6�2
� 3.472
100. 1 �1
2�
1
4�
1
8� . . . �
1
128�
1
20�
1
21�
1
22�
1
23� . . . �
1
27�
7
n�0��
1
2�n
0.664
102.1
1 � 3�
1
2 � 4�
1
3 � 5� . . . �
1
10 � 12�
10
k�1
1
k�k � 2� 0.663
87. 4
i�1
��i � 1�2 � �i � 1�3� � ��0�2 � �2�3� � ��1�2 � �3�3� � ��2�2 � �4�3� � ��3�2 � �5�3� � 238
89. 4
i�1
2i � 21 � 22 � 23 � 24 � 30
88. 5
k�2
�k � 1��k � 3� � �2 � 1��2 � 3� � �3 � 1��3 � 3� � �4 � 1��4 � 3� � �5 � 1��5 � 3� � 14
91. 6
j�1
�24 � 3j� � 81 92. 10
j�1
3
j � 1 6.06 93.
4
k�0
��1�k
k � 1�
47
6094.
4
k�0
��1�k
k!�
3
8� 0.375
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698 Chapter 8 Sequences, Series, and Probability
106. 5
i�1 2�1
3�i
�242243
0.9959
108. 4
n�1 8��
14�
n
��5132
�1.59375
107. 3
n�14��1
2�n
� �1.5 � �32
109. (a)
�33335000
� 0.6666
4
i�16� 1
10�i� 6� 1
10� � 6� 110�
2� 6� 1
10�3
� 6� 110�
4(b)
�23
� 0.666 . . .
� 6�0.111 . . .�
�
i�16� 1
10�i� 6�0.1 � 0.01 � 0.001 � . . .�
110. (a)
�11112500
� 0.4444
4
k�14� 1
10�k
� 4� 110� � 4� 1
10�2
� 4� 110�
3� 4� 1
10�4
(b)
�49
� 0.444 . . .
� 4�0.111 . . .�
�
k�14� 1
10�k� 4�0.1 � 0.01 � 0.001 � . . .�
111. (a)
�1111
10,000
� 0.1111
4
k�1� 1
10�k
�110
�1
100�
11000
�1
10,000(b)
�19
� 0.111 . . .
�
k�1� 1
10�k� 0.1 � 0.01 � 0.001 � . . .
112. (a)
�11115000
� 0.2222
� 2�0.1111�
4
i�12� 1
10�i� 2�0.1� � 2�0.01� � 2�0.001� � 2�0.0001� (b)
�29
� 0.2222 . . .
�
i�12� 1
10�i�2�0.1� � 2�0.01� � 2�0.001� � . . .
104.1
2�
2
4�
6
8�
24
16�
120
32�
720
64�
6
k�1
k!
2k� 18.25 105.
4
i�15�1
2�i
� 4.6875 �7516
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Section 8.1 Sequences and Series 699
113.
(a)
(b) A40 $6741.74
A8 $5307.99
A7 $5268.48 A6 $5229.26
A5 $5190.33 A4 $5151.70
A3 $5113.35 A2 $5075.28
A1 � 5000�1 �0.03
4 �1
� $5037.50
An � 5000�1 �0.03
4 �n
, n � 1, 2, 3, . . . 114. (a)
(b)
(c) A240 � 100�101���1.01�240 � 1� $99,914.79
A60 � 100�101���1.01�60 � 1� $8248.64
A6 � 100�101���1.01�6 � 1� $621.35
A5 � 100�101���1.01�5 � 1� $515.20
A4 � 100�101���1.01�4 � 1� $410.10
A3 � 100�101���1.01�3 � 1� $306.04
A2 � 100�101���1.01�2 � 1� � $203.01
A1 � 100�101���1.01�1 � 1� � $101.00
115. (a) (year 2008)
(b) (2009)
(2010)
(2011)
(Answers will vary slightly.)
(c) The population approaches 2000 trout because0.75�2000� � 500 � 2000.
p3 � 0.75p2 � 500 3477
p2 � 0.75p1 � 500 3969
p1 � 0.75p0 � 500 � 4625
pn � 0.75pn�1 � 500
p0 � 5500 116. (a) (year 2010)
(b)
(c) The number of trees approaches 7500 because0.9�7500� � 750 � 7500.
t4 9140
t3 9323
t2 � 0.9t1 � 750 � 9525
t1 � 0.9t0 � 750 � 9750
tn � 0.9tn�1 � 750
t0 � 10,000
117. (a) (end of January)
(c) After 50 deposits, a49 $2832.26.
an � �1 �0.0612 �an�1 � 50 � 1.005an�1 � 50
a0 � 50 (b) (end of February)
(end of December)
After one year, the IRA has $616.78.
a11 � 616.78
a10 � 563.96a9 � 511.40
a8 � 459.11a7 � 407.07
a6 � 355.29a5 � 303.78
a4 � 252.51a3 � 201.51
a2 � 1.005 a1 � 50 � 150.75
a1 � 1.005 a0 � 50 � 100.25
118. Monthly interest rate is
(a)
(b)
bn � bn�1�1.0075� � 1206.94
b1 � b0�1.0075� � 1206.94 � 149,918.06
b0 � 150,000
0.0912
� 0.0075.
n 0 60 120 180 240 300 360
150,000 143,819.75 134,143.44 118,993.43 95,273.35 58,135.27 �11.12bn
(c) Total amount:
(d) Total interest: 434,487.28 � $150,000 � $284,487.28
1206.94 � 360 � 11.12 � $434.487.28
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700 Chapter 8 Sequences, Series, and Probability
119. (a)
(b) For 2010, and
For 2015, and
(c) Answers will vary.
(d) and So, the averagehourly wage reaches $12 in 2008.
r19 � 12.12.r18 � 11.97
r25 $12.64.n � 25
r20 $12.25.n � 20
8 159
12 120. (a)
(b) For 2005, and thousand.
For 2010, and thousand.
For 2015, and thousand.
(c) Answers will vary.
S25 119,349n � 25
S20 39,671n � 20
S15 10,876n � 15
7 147000
8000
121. (a)
(b) Linear:
Quadratic:
Coefficient of determination for linear model:0.98656
Coefficient of determination for quadraticmodel: 0.99919
(c)
(d) The quadratic model is better.
The quadratic model is better because its coefficient of determination is closer to 1.
(e) For 2010, and million.
For 2015, and million.
(f) when or in 2010.n 20.8,Rn � 1000
R25 1494.5n � 25
R20 912.8n � 20
8 18100
600
8 18100
600
Rn � 3.088n2 � 22.62n � 130.0
Rn � 54.58n � 336.3
8 18100
600 122. (a)
(b) Linear:
Quadratic:
Coefficient of determination for linear model:0.98201
Coefficient of determination for quadraticmodel: 0.98759
(c)
(d) The quadratic model is better.
The quadratic model is better because its coefficient of determination is closer to 1.
(e) For 2010, and million.
For 2015, and million.
(f) when So, sales willreach 20 billion in 2012.
S23 20.7.S22 19.9
S25 22.3n � 25
S20 18.4n � 20
4 180
20
4 180
20
Sn � 0.012n2 � 0.24n � 8.8
Sn � 0.50n � 7.6
4 180
20
123. True 124. True
4
j�1 2 j � 21 � 22 � 23 � 24 �
6
j�3 2 j�2
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Section 8.1 Sequences and Series 701
125.
b5 �138a5 � 5 � 3 � 8
b4 �85a4 � 3 � 2 � 5
b3 �53a3 � 2 � 1 � 3
b2 �32a2 � 1 � 1 � 2
b1 �21 � 2a1 � 1
b0 �11 � 1a0 � 1
a0 � 1, a1 � 1, ak�2 � ak�1 � ak
a11 � 89 � 55 � 144
a10 � 55 � 34 � 89
b9 �8955a9 � 34 � 21 � 55
b8 �5534a8 � 21 � 13 � 34
b7 �3421a7 � 13 � 8 � 21
b6 �2113a6 � 8 � 5 � 13
126.
� 1 �an�1
an
� 1 �1an
an�1
� 1 �1
bn�1
bn �an�1
an
�an � an�1
an
127.
a5 � 5a4 � 3,
a3 � 2a2 � 1,
a1 ��1 � �5�1 � �1 � �5�1
21�5� 1
an ��1 � �5�n
� �1 � �5�n
2n�5
128. These are the first five terms of the Fibonaccisequence.
129.
an�2 ��1 � �5�n�2
� �1 � �5�n�2
2n�2�5
an�1 ��1 � �5�n�1
� �1 � �5�n�1
2n�1�5
130.
Yes, this is the recursive formula for the Fibonacci sequence.
� an�2
��1 � �5�n�2
� �1 � �5�n�2
2n�2�5
��1 � �5�n�1 � �5�2
� �1 � �5�n�1 � �5�2
2n�2�5
��1 � �5�n�2�1 � �5� � 4� � �1 � �5�n�2�1 � �5� � 4�
2n�2�5
�2�1 � �5�n�1
� 2�1 � �5�n�1� 4�1 � �5�n
� 4�1 � �5�n
2n�2�5
an�1 � an ��1 � �5�n�1
� �1 � �5�n�1
2n�1�5�
�1 � �5�n� �1 � �5�n
2n�5
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702 Chapter 8 Sequences, Series, and Probability
131.
a5 �x5
5!�
x5
120
a4 �x4
4!�
x4
24
a3 �x3
3!�
x3
6
a2 �x2
2!�
x2
2
a1 �x1
� x
an �xn
n!
135.
a5 ��x10
10!�
�x10
3,628,800
a4 �x8
8!�
x8
40,320
a3 ��x6
6!�
�x6
720
a2 �x4
4!�
x4
24
a1 ��x2
2
an ���1�nx2n
�2n�!
132.
a5 �x2
25
a4 �x2
16
a3 �x2
9
a2 �x2
4
a1 �x2
1
an �x2
n2 133.
a5 ��x11
11
a4 �x9
9
a3 � �x7
7
a2 �x5
5
a1 ��x3
3
an ���1�nx2n�1
2n � 1
136.
a5 ��x11
11!�
�x11
39,916,800
a4 �x9
9!�
x9
362,880
a3 � �x7
7!�
�x7
5040
a2 �x5
5!�
x5
120
a1 ��x3
3!�
�x3
6
an ���1�nx2n�1
�2n � 1�!134.
a5 ��x6
6
a4 �x5
5
a3 ��x4
4
a2 �x3
3
a1 ��x2
2
an ���1�n xn�1
n � 1
137.
a5 ��x5
5!� �
x5
120
a4 �x4
4!�
x4
24
a3 ��x3
3!�
�x3
6
a2 �x2
2
a1 � �x
an ���1�n xn
n!138.
a5 ��x6
6!� �
x6
720
a4 �x5
5!�
x5
120
a3 ��x4
4!� �
x4
24
a2 �x3
3!�
x3
6
a1 ��x2
2
an ���1�n xn�1
�n � 1�!139.
a5 ��x � 1�5
120
a4 � ��x � 1�4
24
a3 ��x � 1�3
6
a2 ���x � 1�2
2
a1 � x � 1
an ���1�n�1�x � 1�n
n!
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Section 8.1 Sequences and Series 703
140.
a5 ���x � 1�5
720
a4 ��x � 1�4
120
a3 ���x � 1�3
24
a2 ��x � 1�2
6
a1 ���x � 1�
2
an ���1�n �x � 1�n
�n � 1�!
141.
th partial sum �12
�1
2n � 2� �1
2�
14� � �1
4�
16� � . . . � � 1
2n�
12n � 2�n
a5 �110
�112
�160
a4 �18
�110
�140
a3 �16
�18
�124
a2 �14
�16
�112
a1 �12
�14
�14
an �12n
�1
2n � 2
142.
th partial sum � 1 �1
n � 1� �1
1�
12� � �1
2�
13� � . . . � �1
n�
1n � 1�n
a5 �15
�16
�130
a4 �14
�15
�120
a3 �13
�14
�112
a2 �12
�13
�16
a1 � 1 �12
�12
an �1n
�1
n � 1
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704 Chapter 8 Sequences, Series, and Probability
143.
th partial sum �12
�1
n � 2� �1
2�
13� � �1
3�
14� � . . . � � 1
n � 1�
1n � 2�n
a5 �16
�17
�142
a4 �15
�16
�130
a3 �14
�15
�120
a2 �13
�14
�112
a1 �12
�13
�16
an �1
n � 1�
1n � 2
144.
th partial sum
� �1 �12� � � 1
n � 1�
1n � 2� �
32
�1
n � 1�
1n � 2
� �1 �13� � �1
2�
14� � �1
3�
15� � . . . � � 1
n � 1�
1n � 1� � �1
n�
1n � 2�n
a5 �15
�17
�235
a4 �14
�16
�112
a3 �13
�15
�215
a2 �12
�14
�14
a1 � 1 �13
�23
an �1n
�1
n � 2
145.
� ln�n!�
� ln�2 � 3 . . . n�
nth partial sum � ln 2 � ln 3 � . . . � ln n
a5 � ln 5
a4 � ln 4
a3 � ln 3
a2 � ln 2
a1 � ln 1 � 0
an � ln n
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Section 8.2 Arithmetic Sequences and Partial Sums 705
146.
� n � ln��n � 1�!�
� n � ln�2 � 3 . . . �n � 1��
� n � �ln 2 � ln 3 � . . . � ln�n � 1��
nth partial sum � �1 � ln 2� � �1 � ln 3� � . . . � �1 � ln�n � 1��
a5 � 1 � ln 6
a4 � 1 � ln 5
a3 � 1 � ln 4
a2 � 1 � ln 3
a1 � 1 � ln 2
an � 1 � ln�n � 1�
149. (a)
(b)
(c)
(d) BA � �161013
314722
423125�
AB � ��2
41
74223
�164548�
2B � 3A � �8
�12�3
17�13�15
�14�9
�10�
A � B � ��3
41
�744
413�
147. (a)
(b)
(c)
(d) BA � � 027
618�
AB � �1818
90�
2B � 3A � ��223
�7�18�
A � B � � 8�3
17� 148. (a)
(b)
(c)
(d) BA � �4836
�72122�
AB � �5648
�43114�
2B � 3A � ��3028
�454�
A � B � � 10�12
19�5�
150. (a)
(b)
(c)
(d) BA � �2021
415
�6
8�4
6�
AB � �121
�6
021
�1
�828�
2B � 3A � �3
�9�2
�4�1
3
0�10�5�
A � B � ��1
21
00
�1
041�
Section 8.2 Arithmetic Sequences and Partial Sums
■ You should be able to recognize an arithmetic sequence, find its common difference, and find its th term.
■ You should be able to find the th partial sum of an arithmetic sequence with common difference usingthe formula
Sn �n
2�a1 � an �.
dn
n
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706 Chapter 8 Sequences, Series, and Probability
1. 10, 8, 6, 4, 2, . . .
Arithmetic sequence, d � �2
3.
Arithmetic sequence, d � �12
3, 52, 2, 32, 1, . . .
5.
Arithmetic sequence, d � 8
�24, �16, �8, 0, 8
7.
Arithmetic sequence, d � 0.6
3.7, 4.3, 4.9, 5.5, 6.1, . . . 9.
21, 34, 47, 60, 73
Arithmetic sequence, d � 13
an � 8 � 13n
11.
Not an arithmetic sequence
1
2,
1
3,
1
4,
1
5,
1
6
an �1
n � 1
2.
Arithmetic sequence, d � 5
4, 9, 14, 19, 24, . . .
4.
Not an arithmetic sequence
13, 23, 43, 83, 16
3 , . . . 6. ln 1, ln 2, ln 3, ln 4, ln 5, . . .
Not an arithmetic sequence
8.
Not an arithmetic sequence
12, 22, 32, 42, 52, . . .
10.
3, 6, 11, 20, 37
Not an arithmetic sequence
an � 2n � n 12.
1, 5, 9, 13, 17
Arithmetic sequence, d � 4
an � 1 � �n � 1� 4
13.
143, 136, 129, 122, 115
Arithmetic sequence, d � �7
an � 150 � 7n 15.
1, 5, 1, 5, 1
Not an arithmetic sequence
an � 3 � 2��1�n
17.
an � a1 � �n � 1�d � 1 � �n � 1��3� � 3n � 2
a1 � 1, d � 3
14.
1, 2, 4, 8, 16
Not an arithmetic sequence
an � 2n�1
16.
Arithmetic sequence, d � �4
a5 � �41
a4 � �37
a3 � �33
a2 � �29
a1 � �25
an � 3 � 4�n � 6� � �21 � 4n
18.
an � a1 � �n � 1�d � 15 � �n � 1� 4 � 11 � 4n
a1 � 15, d � 4
20.
an � a1 � �n � 1�d � �n � 1� ��23� �
23 �
23n
a1 � 0, d � �23
19.
� 100 � �n � 1���8� � 108 � 8n
an � a1 � �n � 1�d
a1 � 100, d � �8
21.
an � a1 � �n � 1�d � 4 � �n � 1���52� �
132 �
52n
4, 32, �1, �72, . . . , d � �
52
Vocabulary Check
1. arithmetic, common 2. 3. th partial sumnan � dn � c
23.
an � a1 � �n � 1�d � 5 � �n � 1��103 � �
103 n �
53
a4 � a1 � 3d ⇒ 15 � 5 � 3d ⇒ d �103
a1 � 5, a4 � 1522.
� 15 � 5n
an � a1 � �n � 1�d � 10 � �n � 1� ��5�
10, 5, 0, �5, �10, . . . , d � �5
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Section 8.2 Arithmetic Sequences and Partial Sums 707
24.
an � �4 � �n � 1�5 � �9 � 5n
d � 5
16 � �4 � 4d
an � a1 � �n � 1� d
a1 � �4, a5 � 16 25.
� 100 � �n � 1���3� � 103 � 3n
an � a1 � �n � 1�d
a1 � a3 � 2d ⇒ a1 � 94 � 2��3� � 100
a6 � a3 � 3d ⇒ 85 � 94 � 3d ⇒ d � �3
a3 � 94, a6 � 85
27.
a5 � 23 � 6 � 29
a4 � 17 � 6 � 23
a3 � 11 � 6 � 17
a2 � 5 � 6 � 11
a1 � 5
a1 � 5, d � 6
29.
a5 � �46 � 12 � �58
a4 � �34 � 12 � �46
a3 � �22 � 12 � �34
a2 � �10 � 12 � �22
a1 � �10
a1 � �10, d � �12
26.
� 265 � 15n
an � a1 � �n � 1� d � 250 � �n � 1� ��15�
a1 � a5 � 4d ⇒ a1 � 190 � 4��15� � 250
a10 � a5 � 5d ⇒ 115 � 190 � 5d ⇒ d � �15
a5 � 190, a10 � 115
28.
a5 �114 �
34 �
84 � 2
a4 �72 �
34 �
114
a3 �174 �
34 �
144 �
72
a2 � 5 �34 �
174
a1 � 5
a1 � 5, d � �34
30.
Answer:
a5 � 16 � 5 � 21
a4 � 11 � 5 � 16
a3 � 6 � 5 � 11
a2 � 1 � 5 � 6
a1 � 1
a1 � 1, d � 5
46 � a10 � a1 � �n � 1�d � a1 � 9d
16 � a4 � a1 � �n � 1�d � a1 � 3d
a4 � 16, a10 � 46 31.
Answer:
a5 � 10 � 4 � 14
a4 � 6 � 4 � 10
a3 � 2 � 4 � 6
a2 � �2 � 4 � 2
a1 � �2
d � 4, a1 � �2
42 � a12 � a1 � �n � 1�d � a1 � 11d
26 � a8 � a1 � �n � 1�d � a1 � 7d
a8 � 26, a12 � 42
32.
a5 � �24 � 7 � �31
a4 � �17 � 7 � �24
a3 � �10 � 7 � �17
a2 � �3 � 7 � �10
a6 � a1 � 5d ⇒ �38 � a1 � 5��7� ⇒ a1 � �3
�73 � �38 � 5d ⇒ d � �7
a11 � a6 � 5d 33.
a5 � 17.275 � 1.725 � 15.55
a4 � 19 � 1.725 � 17.275
a3 � 19
a2 � a1 � 1.725 � 20.725
⇒ a1 � 22.45
a3 � a1 � 2d ⇒ 19 � a1 � 2��1.725�
�1.7 � 19 � 12d ⇒ d � �1.725
a15 � a3 � 12d
a3 � 19, a15 � �1.7
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708 Chapter 8 Sequences, Series, and Probability
34.
a5 � 13.5 � 2.5 � 16
a4 � 11 � 2.5 � 13.5
a3 � 8.5 � 2.5 � 11
a2 � 6 � 2.5 � 8.5
a5 � a1 � 4d ⇒ 16 � a1 � 4�2.5� ⇒ a1 � 6
38.5 � 16 � 9d ⇒ d � 2.5
a14 � a5 � 9d
36.
an � �10n � 210
d � �10
a5 � 170 � 10 � 160
a4 � 180 � 10 � 170
a3 � 190 � 10 � 180
a2 � 200 � 10 � 190
ak�1 � ak � 10
a1 � 200
38.
an � 4.0 � 2.5n
d � �2.5
a5 � �6.0 � 2.5 � �8.5
a4 � �3.5 � 2.5 � �6.0
a3 � � 1.0 � 2.5 � �3.5
a2 � 1.5 � 2.5 � �1.0
a1 � 1.5, ak�1 � ak � 2.5 40.
� 3 � 8�10� � 83
a9 � a1 � 8d
13 � 3 � d ⇒ d � 10
a2 � a1 � d
35.
d � 4, an � 11 � 4n
a5 � 27 � 4 � 31
a4 � 23 � 4 � 27
a3 � 19 � 4 � 23
a2 � a1 � 4 � 15 � 4 � 19
a1 � 15, ak�1 � ak � 4
37.
an �710 �
110n
d � �110
a5 � �110 �
310 �
15
a4 � �110 �
25 �
310
a3 � �110 �
12 �
410 �
25
a2 � �110 �
35 �
510 �
12
a1 �35, ak�1 � �
110 � ak
39.
a10 � a1 � 9d � 5 � 9�6� � 59
a1 � 5, a2 � 11 ⇒ d � 6
41.
a7 � a1 � 6d � 4.2 � 6�2.4� � 18.6
a1 � 4.2, a2 � 6.6 ⇒ d � 2.4 42.
a8 � a1 � 7d � �0.7 � 7��13.1� � �92.4
d � a2 � a1 � �13.8 � ��0.7� � �13.1
44.
−6
0 11
16an � �5 � 2n43.
00 11
16an � 15 �32n
45.
00 11
10an � 0.5n � 4 46.
−9
0 11
3an � �0.9n � 2
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Section 8.2 Arithmetic Sequences and Partial Sums 709
48. an � 17 � 3n
n 1 2 3 4 5 6 7 8 9 10
20 23 26 29 32 35 38 41 44 47an
47. an � 4n � 5
49. an � 20 �34n
n 1 2 3 4 5 6 7 8 9 10
3 7 11 15 19 23 27 31 35�1an
n 1 2 3 4 5 6 7 8 9 10
19.25 18.5 17.75 17 16.25 15.5 14.75 14 13.25 12.5an
50. an �45 n � 12
n 1 2 3 4 5 6 7 8 9 10
12.8 13.6 14.4 15.2 16 16.8 17.6 18.4 19.2 20an
51. an � 1.5 � 0.05n
n 1 2 3 4 5 6 7 8 9 10
1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2.0an
52. an � 8 � 12.5n
n 1 2 3 4 5 6 7 8 9 10
�117�104.5�92�79.5�67�54.5�42�29.5�17�4.5an
53. S10 �102 �2 � 20� � 110 54. S7 �
72�1 � 19� � 70 55. S5 �
52��1 � ��9�� � �25
56. S6 �62��5 � 5� � 0 57. S50 �
502 �2 � 100� � 2550 58.
� 10,000
�100
n�1
�2n � 1� �1002 �1 � 199�
a1 � 1, a100 � 199, n � 100
59. S131 �1312 ��100 � 30� � �4585 60.
�60
i�0
�i � 10� �612 ��10 � 50� � 1220
a1 � �10, a61 � 50, n � 61
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710 Chapter 8 Sequences, Series, and Probability
68.
�100
n�1
2n �1002 �2 � 200� � 10,100
a1 � 2, a100 � 200, n � 100
an � 2n
63.
S10 �102 �a1 � a10 � � 5�0.5 � 7.7� � 41
a10 � a1 � 9d � 0.5 � 9�0.8� � 7.7
a1 � 0.5, a2 � 1.3 ⇒ d � 0.8 64.
S12 �122 �4.2 � 1.3� � 17.4
a12 � 4.2 � 11��0.5� � �1.3
n � 12
d � �0.5
a1 � 4.2
4.2, 3.7, 3.2, 2.7, . . . n � 12
66.
S100 �1002 �15 � 307� � 16,100
a1 � 15, a100 � 307, n � 10065.
S25 �252 �a1 � a25� � 12.5�100 � 220� � 4000
a1 � 100, a25 � 220
67.
�50
n�1
n �502 �1 � 50� � 1275
a1 � 1, a50 � 50, n � 50
61.
S10 �n2
�a1 � a10� �102
�8 � 116� � 620
a10 � a1 � 9d � 8 � 9�12� � 116
a1 � 8, a2 � 20 ⇒ d � 12
8, 20, 32, 44, . . . n � 10 62.
S50 �502
��6 � 190� � 4600
a50 � �6 � 49�4� � 190
a1 � �6, d � 4, n � 50
�6, �2, 2, 6, . . .
69.
�100
n�1
5n �1002 �5 � 500� � 25,250
a1 � 5, a100 � 500, n � 100 70.
�100
n�51
7n �502 �357 � 700� � 26,425
a51 � 357, a100 � 700
an � 7n
72.
� 3775 � 1275 � 2500
�100
n�51
n � �50
n�1
n �502 �51 � 100� �
502 �1 � 50�71.
� 410 � 55 � 355
�30
n�11
n � �10
n�1
n �202 �11 � 30� �
102 �1 � 10�
73. �500
n�1 �n � 8� �
5002 �9 � 508� � 129,250 74.
�250
n�1
�1000 � n� �2502 �999 � 750� � 218,625
a1 � 999, a250 � 750, n � 250
an � 1000 � n
75. �20
n�1�2n � 1� � 440 76. �
50
n�0�50 � 2n� � 0 77. �
100
n�1
n � 12
� 2575
78. �100
n�0
4 � n4
� �1161.5 79. �60
i�1�250 �
25
i � 14,268 80. �200
j�1�10.5 � 0.025j� � 2602.5
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Section 8.2 Arithmetic Sequences and Partial Sums 711
82.
S10 �102 �15 � 24� � 195 logs
a1 � 15, a10 � 24, d � 1, n � 10
84.
S10 �102 �4.9 � 93.1� � 490 meters
a10 � 9.8�10� � 4.9 � 93.1
an � 9.8n � 4.9
a1 � 4.9 � 9.8�1� � c ⇒ c � �4.9
a4 � 34.3 ⇒ d � 9.8
a1 � 4.9, a2 � 14.7, a3 � 24.5,
81.
S18 �182 �14 � 31� � 405 bricks
a1 � 14, a18 � 31
83.
S5 �52 �20,000 � 40,000� � 150,000
a5 � 20,000 � 4�5000� � 40,000
d � 5000
a2 � 20,000 � 5000 � 25,000
a1 � 20,000
85. (a)
(b)
The model is a good fit.
(c) Total billion
(d) For 2005, and
For 2012, and
Total billion
Answers will vary.
�82�19.35 � 25.72� � $180.3
S22 � 25.72.n � 22
S15 � 19.35.n � 15
�82�12.1 � 18.4� � $122
Sn � 0.91n � 5.7
Year 1997 1998 1999 2000 2001 2002 2003 2004
Sales 12.1 13.0 13.9 14.8 15.7 16.6 17.5 18.4(Billions of $)
86. (a) corresponds to 1995.
(b)
an � 13.5n � 324, n � 5 (c) thousand
Adding the table entries,
thousand.
(d) For 2004 to 2014,
thousand.
Answers will vary.
S �112 �513 � 648� 6386
398 � . . . � 512 � 4012
S �92�392 � 500� 4014
Year Model
1995 392
1996 405
1997 419
1998 432
1999 446
2000 459
2001 473
2002 486
2003 500
87. True. Given you know Thus, an � a1 � �n � 1�d.
d � a2 � a1.a1 and a2 , 88. False. You need to know how many terms are in the sequence.
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712 Chapter 8 Sequences, Series, and Probability
90.
a10 � 44ya5 � 19y
a9 � 39ya4 � 14y
a8 � 34ya3 � 9y
a7 � 29ya2 � �y � 5y � 4y
a6 � 24ya1 � �y89.
a10 � 19xa5 � 9x
a9 � 17xa4 � 7x
a8 � 15xa3 � 3x � 2x � 5x
a7 � 13xa2 � x � 2x � 3x
a6 � 11xa1 � x
94. Gauss might have done the following:
Adding:
In general, 1 � 2 � . . . � n �n�n � 1�
2.
100�101� � 2x ⇒ x �100�101�
2� 5050
101 � 101 � . . . � 101 � 101 � 2x
100 � 99 � . . . � 2 � 1 � x
1 � 2 � 3 � . . . � 99 � 100 � x
95. S �n�n � 1�
2�
200�201�2
� 20,100 96.
� 2�100�101�2 � 10,100
� 2�1 � 2 � . . . � 100�
S � 2 � 4 � 6 � . . . � 200
97.
� 5151 � 2550 � 2601
�101�102�
2� 2�50�51�
2 � �1 � 2 � 3 � . . . � 101� � �2 � 4 � . . . � 100�
S � 1 � 3 � 5 � . . . � 101
98.
� 200�101� � 20,200
� 4 �100�101�
2
4 � 8 � . . . � 400 � 4�1 � 2 � . . . � 100�
91.
a1 � 4
20a1 � 80
10�2a1 � 57� � 650
�202
�a1 � �a1 � 57�� � 650
S �n2
�a1 � a20�
a20 � a1 � 19�3� � a1 � 57 92.
�n2
�a1 � a2� � 5n
�n2
�a1 � a2 � 10�
S �n2
��a1 � 5� � �an � 5��
93. (a)
(b) 17, 23, 29, 35, 41, 47, 53, 57
(c) Not arithmetic
(d) 4, 7.5, 11, 14.5, 18, 21.5, 25, 28.5
(e) Not arithmetic
an�1 � an � 3.5, a1 � 4
an�1 � an � 6, a1 � 17
an�1 � an � 3, a1 � �7
�7, �4, �1, 2, 5, 8, 11
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Section 8.3 Geometric Sequences and Series 713
99. row reduces to
Answer: �1, 5, �1�
�100
010
001
�
�
�
15
�1�.�236
�12
�5
7�4
1
�
� �
�1017
�20�
100. row reduces to
Answer: �2, �6, 3�
�100
010
001
�
�
�
2�6
3�.��1
58
4�3
2
101
�3
�
�
�
431
�5�
101.
square unitsArea �12 �30� � 15
�042 0�3
6
111� � 30 102.
Area square units�12�40� � 20
��153
218
111� � 40
103. Answers will vary.
Section 8.3 Geometric Sequences and Series
■ You should be able to identify a geometric sequence, find its common ratio, and find the th term.
■ You should be able to find the th partial sum of a geometric sequence with common ratio usingthe formula.
■ You should know that if then
��
n�1
a1rn�1 �
a1
1 � r.
�r� < 1,
Sn � a1�1 � rn
1 � r �
rn
n
1. 5, 15, 45, 135, . . .
Geometric sequence
r � 3
3. 6, 18, 30, 42, . . .
Not a geometric sequence
Note: It is an arithmeticsequence with d � 12.��
2.
Geometric sequence
r �123 � 4
3, 12, 48, 192, . . .
Vocabulary Check
1. geometric, common 2. 3.
4. geometric series 5. S � ��
i�0a1r
i �a1
1 � r
Sn � �n
i�1a1r
i�1 � a1�1 � rn
1 � r �an � a1rn�1
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714 Chapter 8 Sequences, Series, and Probability
12.
a5 � 32�2� � 64
a4 � 16�2� � 32
a3 � 8�2� � 16
a2 � 4�2� � 8
a1 � 4, r � 211.
a5 � 162�3� � 486
a4 � 54�3� � 162
a3 � 18�3� � 54
a2 � 6�3� � 18
a1 � 6, r � 3 13.
a5 �18�1
2� �116
a4 �14�1
2� �18
a3 �12�1
2� �14
a2 � 1�12� �
12
a1 � 1
a1 � 1, r �12
15.
a5 � �� 1200��� 1
10� �1
2000
a4 �120�� 1
10� � �1
200
a3 � ��12��� 1
10� �120
a2 � 5�� 110� � �
12
a1 � 5
a1 � 5, r � �110
17.
a5 � �e3��e� � e4
a4 � �e2��e� � e3
a3 � �e��e� � e2
a2 � 1�e� � e
a1 � 1
a1 � 1, r � e 19.
r �12, an � 64�1
2�n�1� 128�1
2�n
a5 �12 �8� � 4
a4 �12 �16� � 8
a3 �12 �32� � 16
a2 �12 �64� � 32
a1 � 64
a1 � 64, ak�1 �12 ak
21.
an � �92�2n � 9�2n�1�
r � 2
a5 � 2�72� � 144
a4 � 2�36� � 72
a3 � 2�18� � 36
a2 � 2�9� � 18
a1 � 9, ak�1 � 2ak
14.
a5 �227 �1
3� �281
a4 �29 �1
3� �227
a3 �23 �1
3� �29
a2 � 2�13� �
23
a1 � 2, r �13 16.
a5 � 6��14�4
�3
128
a4 � 6��14�3
� �332
a3 � 6��14�2
�38
a2 � 6��14�1
� �32
a1 � 6
a1 � 6, r � �14
18.
a5 � 123�3� � 36
a4 � 12�3� � 123
a3 � 43�3� � 12
a2 � 43
a1 � 4, r � 3
20.
r �13, an � 243� 1
3�n
a5 �13�3� � 1
a4 �13�9� � 3
a3 �13�27� � 9
a2 �13�81� � 27
a1 � 81
a1 � 81, ak�1 �13ak 22.
r � �3, an � 5��3�n�1
a5 � �135��3� � 405
a4 � 45��3� � �135
a3 � �15��3� � 45
a2 � �3�5� � �15
a1 � 5, ak�1 � �3ak
4.
Geometric sequence
r � �2
1, �2, 4, �8, . . . 5.
Geometric sequence
r � �12
1, �12,
14, �
18, . . . 6. 5, 1, 0.2, 0.04
Geometric sequence
r �15 � 0.2
7.
Geometric sequence
r � 2
18, 14, 12, 1, . . .
8.
Geometric sequence
r � �23
9, �6, 4, �83, . . . 9.
Not a geometric sequence
1, 12, 13, 14, . . . 10.
Not a geometric sequence
15, 27, 39, 4
11, . . .
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Section 8.3 Geometric Sequences and Series 715
23.
r � �3
2, an � 6��3
2�n�1
� �4��3
2�n
a5 � �3
2 ��81
4 � �243
8
a4 � �3
2 �27
2 � � �81
4
a3 � �3
2��9� �
27
2
a2 � �3
2�6� � �9
a1 � 6
a1 � 6, ak�1 � �3
2ak 24.
r ��2
3, an � 30��2
3 �n�1
a5 ��2
3 ��80
9 � �160
27
a4 ��2
3 �40
3 � ��80
9
a3 ��2
3��20� �
40
3
a2 ��2
3a1 �
�2
3�30� � �20
a1 � 30, ak�1 � �2
3ak
28.
a9 � 8��34 �8
�65618192
an � a1rn�1
a1 � 8, r ��34
25.
a10 � a4r6 �
12�
12�
6
�127 �
1128
r �12
r3 �18
4r3 �12
a1r3 � a4
a1 � 4, a4 �12
, n � 10 26.
a8 � a3r5 �
454 �±
32�
5� ±
10,935128
r � ±32
r2 �94
5r2 �454
a1r2 � a3
a1 � 5, a3 �454
, n � 8
27.
a12 � 6��1
3�11
��2
310
1an � a1rn�1
1a1 � 6, r � �1
3, n � 12 29.
a14 � 500�1.02�13 646.8
4an � a1rn�1
4a1 � 500, r � 1.02, n � 14
30.
1051.14a11 � 1000�1.005�10
an � a1rn�1
n � 11
a1 � 1000, r � 1.005, 31.
a6 � a1r5 � 54��1
3 �5
��54
243� �
2
9
a1 ��18
r�
�18
�1�3� 54
a5 � a1r4 � �a1r�r3 � �18r3 �
2
3 ⇒ r � �
1
3
a2 � a1r � �18 ⇒ a1 ��18
r
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716 Chapter 8 Sequences, Series, and Probability
32.
a7 � a5r 2 �6427�±2
3�2�
256243
r � ±23
r2 �49
163 r2 �
6427
a3r2 � a5
a3 �163 , a5 �
6427, n � 7 33. 7, 21, 63
a9 � 7�3�9�1 � 45,927
an � 7�3�n�1
r � 3
34. 3, 36, 432
a7 � 3�12�7�1 � 8,957,952
an � 3�12�n�1
r �363 � 12
35. 5, 30, 180
a10 � 5�6�10�1 � 50,388,480
an � 5�6�n�1
r �305 � 6
36. 4, 8, 16
a22 � 4�2�22�1 � 8,388,608
an � 4�2�n�1
r �84 � 2
37.
−10
0 11
14
an � 12��0.75�n�1
39.
00 11
24
an � 2�1.3�n�1
41.
S4 � 8 � ��4� � 2 � ��1� � 5
S3 � 8 � ��4� � 2 � 6
S2 � 8 � ��4� � 4
S1 � 8
8, �4, 2, �1, 12
38.
00 11
150 40.
−50
0 11
50
42.
S4 � 8 � 12 � 18 � 27 � 65
S3 � 8 � 12 � 18 � 38
S2 � 8 � 12 � 20
S1 � 8
8, 12, 18, 27, 812 , . . .
43. ��
n�116��1
2�n�1n 1 2 3 4 5 6 7 8 9 10
16 24 28 30 31 31.5 31.75 31.875 31.9375 31.96875Sn
44. ��
n�1 4�0.2�n�1 n 1 2 3 4 5 6 7 8 9 10
4 4.8 4.96 4.992 4.9984 4.99968 4.999936 4.9999872 5 5Sn
46.
S9 �1�1 � ��2�9�
1 � ��2�� 171
�9
n�1
��2�n�1 ⇒ a1 � 1, r � �245.
S9 �1�1 � 29�
1 � 2� 511
�9
n�1
2n�1 ⇒ a1 � 1, r � 2
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Section 8.3 Geometric Sequences and Series 717
48.
S6 � 32�1 � �1�4�6�
1 � �1�4��
1365
32
�6
i�1 32�1
4�i�1
⇒ a1 � 32, r �1447.
S7 � 64�1 � ��1�2�7
1 � ��1�2� � �128
3 �1 � ��1
2�7
� � 43
�7
i�1
64��1
2�i�1
⇒ a1 � 64, r � �1
2
49.
� �6�1 � �32�
21� 29,921.31
S21 � 3�1 � �3�2�21
1 � �3�2� �
�20
n�0
3�3
2�n
� �21
n�1
3�3
2�n�1
⇒ a1 � 3, r �3
2
51.
S10 � 8�1 � ��1�4�10
1 � ��1�4� � �32
5 �1 � ��1
4�10� 6.4
�10
i�1
8��1
4�i�1
⇒ a1 � 8, r � �1
4
50.
S16 � 2�1 � �4�3�16
1 � �4�3� � 592.65
�15
n�0 2�4
3�n
� �16
n�1 2�4
3�n�1
⇒ a1 � 2, r �43
52.
S10 � 5�1 � ��1�3�10
1 � ��1�3� � 3.75
�10
i�1
5��1
3�i�1
⇒ a1 � 5, r � �1
3
53.
S6 � 300�1 � �1.06�6
1 � 1.06 � 2092.60
�5
n�0
300�1.06�n � �6
n�1
300�1.06�n�1 ⇒ a1 � 300, r � 1.06
54.
S7 � 500�1 � �1.04�7
1 � 1.04 � 3949.15
�6
n�0
500�1.04�n � �7
n�1
500�1.04�n�1 ⇒ a1 � 500, r � 1.04
55.
Thus, the sum can be written as �7
n�1
5�3�n�1.
r � 3 and 3645 � 5�3�n�1 ⇒ n � 7
5 � 15 � 45 � . . . � 3645 56.
and
�8
n�1 7�2�n�1
896 � 7�2�n�1 ⇒ n � 8r � 2
7 � 14 � 28 � . . . � 896
57.
and
�7
n�12��1
4�n�1
12048 � 2��1
4�n�1 ⇒ n � 7r � �
14
2 �12 �
18 � . . . �
12048 58.
and
�6
n�1 15��0.2�n�1
�3
625 � 15��0.2�n�1 ⇒ n � 6r � �0.2
15 � 3 �35 � . . . �
3625
59.
��
n�010�4
5�n
�a1
1 � r�
101 � 4
5
� 50
a1 � 10, r �45
60.
��
n�06�2
3�n
�a1
1 � r�
61 � 2
3
� 18
a1 � 6, r �23
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718 Chapter 8 Sequences, Series, and Probability
61.
��
n�05��1
2�n
�a1
1 � r�
51 � ��1
2� �5
�32� �
103
a1 � 5, r � �12
62.
��
n�09��
23�
n
�a1
1 � r�
91 � ��2
3� �9
�53� �
275
a1 � 9, r � �23
63. does not have a finite sum �73 > 1�.�
�
n�12�7
3�n�164. does not have a finite sum �5
3 > 1�.��
n�18�5
3�n�1
65.
�100089
11.236
��
n�010�0.11�n �
a1
1 � r�
101 � 0.11
�10
0.89
a1 � 10, r � 0.11 66.
9.091
��
n�05�0.45�n �
a1
1 � r�
51 � 0.45
�5
0.55�
10011
a1 � 5, r � 0.45
67.
��31.9
��3019
�1.579
��
n�0�3��0.9�n �
a1
1 � r�
�31 � ��0.9�
a1 � �3, r � �0.9 68.
��101.2
��25
3 �8.333
��
n�0�10��0.2�n �
a1
1 � r�
�101 � ��0.2�
a1 � �10, r � �0.2
69.
�8
1 � 3�4� 32
8 � 6 �9
2�
27
8� . . . � �
�
n�0 8�3
4�n
70.
�9
1 � 2�3�
91�3
� 27
9 � 6 � 4 �83
� . . . � ��
n�0 9�2
3�n
71. 3 � 1 �13
�19
� . . . � ��
n�03��
13�
n
�a1
1 � r�
31 � ��1�3� � 3�3
4� �94
72. �6 � 5 �256
�12536
� . . . � ��
n�0 �6��
56�
n
��6
1 � ��5�6� ��6
11�6�
�3611
�3.2727
73.
�0.36
1 � 0.01�
0.36
0.99�
36
99�
4
11
0.36 � ��
n�0
0.36�0.01�n 74.
�0.297
1 � 0.001�
0.297
0.999�
297
999�
11
37
0.297 � ��
n�0
0 .297�0.001�n
75.
�65
�590
�11390
�65
�0.050.9
�65
�0.05
1 � 0.1
1.25 � 1.2 � ��
n�00.05�0.1�n 76.
� 13
10�
4
45� 1
7
18�
25
18
� 1.3 �0.08
0.9
� 1.3 �0.08
1 � 0.1
1.38 � 1.3 � ��
n�0
0 .08�0.1�n
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Section 8.3 Geometric Sequences and Series 719
77.
(a)
(b)
(c) n � 4: A � 1000�1 �0.03
4 �4�10� 1348.35
n � 2: A � 1000�1 �0.03
2 �2�10� 1346.86
n � 1: A � 1000�1 � 0.03�10 1343.92
A � P�1 �rn�
nt
� 1000�1 �0.03
n �n�10�
(d)
(e) n � 365: A � 1000�1 �0.03365 �
365�10� 1349.84
n � 12: A � 1000�1 �0.0312 �12�10�
1349.35
78.
(a)
(b)
(c)
(d)
(e) n � 365, A � 5563.61
n � 12, A � 5556.46
n � 4, A � 5541.79
n � 2, A � 5520.10
n � 1, A � 2500�1 �0.04
1 �20�1�� 5477.81
A � P�1 �rn�
nt
� 2500�1 �0.04
n �20n
79.
$6480.83
� 100�1.0025� � �1 � 1.002560
1 � 1.0025 �
� 100�1 �0.0312 � �
�1 � �1 � 0.03�12�60 �1 � �1 � 0.03�12�
A � �60
n�1100�1 �
0.0312 �n
80.
$3157.62
� 50�1 �0.0212 � �
�1 � �1 � 0.02�12�60 �1 � �1 � 0.02�12�
A � �60
n�150�1 �
0.0212 �n
81. Let be the total number of deposits.
� P��1 �r
12�12t
� 1��1 �12
r �
� P��1 �r
12�N
� 1��1 �12
r �
� P�12
r� 1���1 � �1 �
r
12�N
�
� P�1 �r
12���12
r ��1 � �1 �r
12�N
�
� P�1 �r
12�1 � �1 �
r
12�N
1 � �1 �r
12�
� P�1 �r
12� �N
n�1�1 �
r
12�n�1
� �1 �r
12��P � P�1 �r
12� � . . . � P�1 �r
12�N�1
�
A � P�1 �r
12� � P�1 �r
12�2
� . . . � P�1 �r
12�N
N � 12t
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720 Chapter 8 Sequences, Series, and Probability
82. Let be the total number of deposits.
�Per�12�ert � 1�
�er�12 � 1�
� Per�12�1 � �er�12�12t �
1 � er�12
� Per�12�1 � �er�12�N �
�1 � er�12�
� �N
n�1
Per�12�n
A � Per�12 � Pe2r�12 � . . . � PeNr�12
N � 12t
84.
(a)
(b) A �75e0.04�12�e0.04�25� � 1�
e0.04�12 � 1 $38,725.81
A � 75��1 �0.0412 �12�25�
� 1��1 �12
0.04� $38,688.25
P � 75, r � 0.04, t � 25
83.
(a) Compounded monthly:
(b) Compounded continuously: A �50e0.07�12�e0.07�20� � 1�
e0.07�12 � 1 $26,263.88
A � 50��1 �0.07
12 �12�20�
� 1��1 �12
0.07� $26,198.27
P � $50, r � 7%, t � 20 years
85.
(a) Compounded monthly:
(b) Compounded continuously: A �100e0.05�12�e0.05�40� � 1�
e0.05�12 � 1 $153,657.02
A � 100��1 �0.0512 �12�40�
� 1��1 �12
0.05� $153,237.86
P � 100, r � 5% � 0.05, t � 40
86.
(a) Compounded monthly:
(b) Compounded continuously: A �20e0.06�12�e0.06�50� � 1�
e0.06�12 � 1 $76,533.16
A � 20��1 �0.06
12 �12�50�� 1��1 �
12
0.06� $76,122.54
P � $20, r � 6%, t � 50 years
87. First shaded area: Second shaded area: Third shaded area:
Total area of shaded region: square units162
4 �
5
n�0 �1
2�n
� 64�1 � �1�2�6
1 � 1�2 � � 128�1 � �12�
6� � 126
162
4�
12
162
4�
14
162
4, etc.
162
4�
12
�162
4162
4
88. 272 �19� � 272 �1
9��89� � 272 �1
9��89�2
� 272 �19��8
9�3� �
3
n�0272�1
9��89�n
�2465
9 273.89 square inches ©H
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Section 8.3 Geometric Sequences and Series 721
89. (a)
(b)
a12 � �0.8�12�70� 4.81 degrees
a6 � �0.8�6�70� 18.35 degrees
an � �0.8�n�70�
�
a1 � 0.8�70� � 56 degrees
a0 � 70 degrees (c)
Thus, the water freezes between 3 and 4 hours, about 3.5 hours.
a4 28.7
a3 35.8
0 140
75
90. (a) Surface area of a sphere is The surface area of the sphere flake is
(b) Volume of a sphere is The volume of the sphere flake is
(c) The surface area is infinite and the volume is finite.
V �4�3�
1 � 1�3� 2�
V �43
��1�3 � 9�43
��13�
3� � 92�43
��19�
3� � . . . �43
� �43
��13� �
43
��13�
2
� . . . � ��
n�0
43
��13�
n
.
43�r2.
S � 4��1�2 � 9�4��13�
2� � 92�4��19�
2� � . . . � 4� � 4� � 4� � . . . � ��
n�14�.
4�r2.
91.
�400
1 � 0.75� $1600
400 � 0.75�400� � �0.75�2�400� � . . . � ��
n�0400�0.75�n
92.
�500
1 � 0.70 $1666.67
500 � 0.70�500� � �0.70�2�500� � . . . � ��
n�0500�0.70�n
93.
�250
1 � 0.80� $1250
250 � 0.80�250� � �0.80�2�250� � . . . � ��
n�0250�0.80�2
94.
�350
1 � 0.75� $1400
350 � 0.75�350� � �0.75�2�350� � . . . � ��
n�0350�0.75�n
95.
�600
1 � 0.725 $2181.82
600 � 0.725�600� � �0.725�2�600� � . . . � ��
n�0600�0.725�n
96.
�450
1 � 0.775� $2000
450 � 0.775�450� � �0.775�2�450� � . . . � ��
n�0450�0.775�n
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.
722 Chapter 8 Sequences, Series, and Probability
97. (a) Option 1:
Option 2:
Option 2 has the larger cumulative amount.
(b) Option 1:
Option 2:
Option 2 has the larger amount.
�1.02�4�32,500� $35,179.05
�1.025�4�30,000� $33,114.39
$169,131.31
32,500 � 1.02�32,500� � . . . � �1.02�4�32,500� � �4
n�032,500�1.02�n
$157,689.86
30,000 � 1.025�30,000� � . . . � �1.025�4�30,000� � �4
n�030,000�1.025�n
98. (a)
(b) (10 years)
(20 years)
(50 years)
(c)
If this trend continues indefinitely, the number of units will be 80,000.
��
i�08000�0.9�i �
80001 � 0.9
� 80,000
�49
i�08000�0.9�i � 79,588 units
�19
i�08000�0.9�i � 70,274 units
�9
i�08000�0.9�i � 52,106 units
8000 � 0.9�8000� � . . . � �0.9�n�1�8000� � �n�1
i�08000�0.9�i
99. (a) Downward:
Upward:
Total distance:
(b) � 5950 feet��
n�0850�0.75�n � �
�
n�0637.5�0.75�n �
8501 � 0.75
�637.5
1 � 0.75
3208.53 � 2406.4 � 5614.93 feet
� �9
n�0637.5�0.75�n 2406.4 feet
0.75�850� � �0.75�2�850� � . . . � �0.75�10�850� � �9
n�0�0.75��850��0.75�n
3208.53 feet
850 � 0.75�850� � �0.75�2�850� � . . . � �0.75�9�850� � �9
n�0850�0.75�n
100. (a) Total distance feet
(b) Total time seconds� 1 � 2� 11 � 0.9
� 1� � 19� 1 � 2 ��
n�1 �0.9�n
� ��
n�0 32�0.81�n � 16 �
321 � 0.81
� 16 152.42
101. False. See definition page 535. 102. False. You multiply the first term by the commonratio raised to the power.�n � 1�
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Section 8.3 Geometric Sequences and Series 723
109. To use the first two terms of a geometric series tofind the th term, first divide the second term bythe first term to obtain the constant ratio. The thterm is the first term multiplied by the commonratio raised to the power.
r �a2
a1, an � a1r
n�1
�n � 1�
nn
107. (a)
The horizontal asymptote of is This corresponds to the sum of the series.
−20
−3 9
28
y � 12.f �x�
��
n�0 6�1
2�n
�6
1 � 1�2� 12
f �x� � 6�1 � 0.5x
1 � 0.5 � (b)
The horizontal asymptote of is This corresponds to the sum of the series.
−6
−6 24
14
y � 10.f �x�
��
n�02�4
5�n
�2
1 � 4�5� 10
f �x� � 2�1 � 0.8x
1 � 0.8 �
103.
a5 �3x3
8 �x2� �
3x4
16
a4 �3x2
4 �x2� �
3x3
8
a3 �3x2 �x
2� �3x2
4
a2 � 3�x2� �
3x2
a1 � 3, r �x2 105.
a9 � 100�ex�8 � 100e8x
an � a1rn�1
a1 � 100, r � ex, n � 9104.
a5 �74x4
2
a4 �73x3
2
a3 �7x2
�7x� �72x2
2
a2 �12
�7x� �7x2
a1 �12
106.
a6 � 4�4x3 �5
�4096243
x5
an � a1rn�1
a1 � 4, r �4x3
, n � 6
108. Given a real number between and 1,which shows that the terms decrease.
�an� � �an�1�r�� < �an�1��1r
110.
an � a1rn�1
a4 � a1r3
a3 � a2r � a1r2
a2 � a1r
a1
111.
Speed �Distance
Time�
400
200�92�2100 �2�2100�
92 45.65 mph
Time �Distance
Speed�
200
50�
200
42� 200� 92
2100� hours
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724 Chapter 8 Sequences, Series, and Probability
112. Your friend mows at the rate of lawns/hour, and your rate is lawns/hour. Together,the time would be
1
�1�4� � �1�6��
1
10�24�
24
10� 2.4 hours.
16
14
113.
� �104 � 2 � �102
det��1�2
2
385
40
�1� � 4��10 � 16� � 1��8 � 6� 114.
� 19 � 32 � �13
det��1�4
0
032
45
�3� � �1��9 � 10� � 4��8 � 0�
115. Answers will vary.
Section 8.4 Mathematical Induction
■ You should be sure that you understand the principle of mathematical induction. If is a statementinvolving the positive integer where is true and the truth of implies the truth of then istrue for all positive integers
■ You should be able to verify (by induction) the formulas for the sums of powers of integers and be able touse these formulas.
■ You should be able to work with finite differences.
n.PnPk�1,PkP1n,
Pn
1.
Pk�1 �5
�k � 1���k � 1� � 1��
5
�k � 1��k � 2�
Pk �5
k�k � 1�2.
�4
�k � 3��k � 4�
Pk�1 �4
��k � 1� � 2� ��k � 1� � 3�
Pk �4
�k � 2��k � 3�
3.
Pk�1 �2k�1
��k � 1� � 1�! �2k�1
�k � 2�!
Pk �2k
�k � 1�! 4.
Pk�1 �2�k�1��1
�k � 1�! �2k
�k � 1�!
Pk �2k�1
k!
Vocabulary Check
1. mathematical induction 2. first
3. arithmetic 4. second
5.
� 1 � 6 � 11 � . . . � �5k � 4� � �5k � 1�
Pk�1 � 1 � 6 � 11 � . . . � �5k � 4� � �5�k � 1� � 4�
Pk � 1 � 6 � 11 � . . . � �5�k � 1� � 4� � �5k � 4�
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Section 8.4 Mathematical Induction 725
7. 1. When
2. Assume that
Then,
Therefore, by mathematical induction, the formula is valid for all positive integer values of n.
� Sk � 2�k � 1� � k�k � 1� � 2�k � 1� � �k � 1��k � 2�.
Sk�1 � 2 � 4 � 6 � 8 � . . . � 2k � 2�k � 1�
Sk � 2 � 4 � 6 � 8 � . . . � 2k � k�k � 1�.
S1 � 2 � 1�1 � 1�.n � 1,
8. 1. When
2. Assume that
Then,
Therefore, by mathematical induction, the formula is valid for all n ≥ 1.
� �k � 1��4�k � 1� � 1�.
� �k � 1��4k � 3�
� 4k2 � 7k � 3
� k�4k � 1� � �8k � 3�
� Sk � �8k � 3�
� 3 � 11 � . . . � �8k � 5� � �8k � 3�
Sk�1 � 3 � 11 � . . . � �8k � 5� � �8�k � 1� � 5�
Sk � 3 � 11 � . . . � �8k � 5� � k�4k � 1�.
n � 1, S1 � 3 � 1�4�1� � 1� � 3
9. 1. When
2. Assume that
Then,
Therefore, by mathematical induction, the formula is valid for all positive integer values of n.
�12
�k � 1��5�k � 1� � 1�.
�12
�5k2 � 11k � 6� �12
�k � 1��5k � 6�
� Sk � �5k � 3� �k2
�5k � 1� � 5k � 3
Sk�1 � 3 � 8 � 13 � . . . � �5k � 2� � �5�k � 1� � 2�
Sk � 3 � 8 � 13 � . . . � �5k � 2� �k2
�5k � 1�.
n � 1, S1 � 3 �12
�5�1� � 1�
6.
� 7 � 13 � 19 � . . . � �6k � 1� � �6k � 7�
Pk�1 � 7 � 13 � 19 � . . . � �6k � 1� � �6�k � 1� � 1�
Pk � 7 � 13 � 19 � . . . � �6�k � 1� � 1� � �6k � 1�
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.
726 Chapter 8 Sequences, Series, and Probability
12. 1. When
2. Assume that
Then,
Therefore, the formula is valid for all positive integer values of n.
� 3k�1 � 1.
� 3 � 3k � 1
� 3k � 1 � 2 � 3k
� Sk � 2 � 3k
Sk�1 � 2�1 � 3 � 32 � 33 � . . . � 3k�1� � 2 � 3k�1�1
Sk � 2�1 � 3 � 32 � 33 � . . . � 3k�1� � 3k � 1.
S1 � 2 � 31 � 1.n � 1,
10. 1. When
2. Assume that
Then,
Therefore, the formula is valid for all positive integer values of n.
�k � 1
2�3�k � 1� � 1�.
��k � 1��3k � 2�
2
�3k2 � 5k � 2
2
�3k2 � k � 6k � 2
2
�k
2�3k � 1� � �3k � 1�
� Sk � �3�k � 1� � 2�
Sk�1 � 1 � 4 � 7 � 10 � . . . � �3k � 2� � �3�k � 1� � 2�
Sk � 1 � 4 � 7 � 10 � . . . � �3k � 2� �k
2�3k � 1�.
S1 � 1 �1
2�3 � 1 � 1�.
n � 1,
11. 1. When
2. Assume that
Then,
Therefore, by mathematical induction, the formula is valid for all positive integer values of n.
� Sk � 2k � 2k � 1 � 2k � 2�2k� � 1 � 2k�1 � 1.
Sk�1 � 1 � 2 � 22 � 23 � . . . � 2k�1 � 2k
Sk � 1 � 2 � 22 � 23 � . . . � 2k�1�2k � 1.
S1 � 1 � 21 � 1.n � 1,
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Section 8.4 Mathematical Induction 727
13. 1. When
2. Assume that
Then,
Therefore, the formula is valid for all positive integer values of n.
� Sk � �k � 1� �k�k � 1�
2�
2�k � 1�2
��k � 1��k � 2�
2.
Sk�1 � 1 � 2 � 3 � 4 � . . . � k � �k � 1�
Sk � 1 � 2 � 3 � 4 � . . . � k �k�k � 1�
2.
n � 1, S1 � 1 �1�1 � 1�
2.
14. 1. When
2. Assume that
Then,
Therefore, the formula is valid for all positive integer values of n.
��k � 1�2�k2 � 4�k � 1��
4�
�k � 1�2�k2 � 4k � 4�4
��k � 1�2�k � 2�2
4.
� Sk � �k � 1�3 �k2�k � 1�2
4� �k � 1�3 �
k2�k � 1�2 � 4�k � 1�3
4
Sk�1 � 13 � 23 � 33 � 43 � . . . � k3 � �k � 1�3
Sk � 13 � 23 � 33 � 43 � . . . � k3 �k2�k � 1�2
4.
n � 1, S1 � 13 � 1 �1�1 � 1�2
4.
15. 1. When
2. Assume that
Then,
Therefore, the formula is valid for all positive integer values of n.
��k � 1��k � 2��2�k � 1� � 1��3�k � 1�2 � 3�k � 1� � 1�
30. �
�k � 1��k � 2��2k � 3��3k2 � 9k � 5�30
��k � 1��6k4 � 39k3 � 91k2 � 89k � 30�
30 �
�k � 1��k�2k � 1��3k2 � 3k � 1� � 30�k � 1�3�30
�k�k � 1��2k � 1��3k2 � 3k � 1� � 30�k � 1�4
30 �
k�k � 1��2k � 1��3k2 � 3k � 1�30
� �k � 1�4
Sk�1 � Sk � �k � 1�4
Sk � �k
i�1
i4 �k�k � 1��2k � 1��3k2 � 3k � 1�
30.
S1 � 14 �1�1 � 1��2 � 1 � 1��3 � 12 � 3 � 1 � 1�
30.
n � 1,
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728 Chapter 8 Sequences, Series, and Probability
17. 1. When
2. Assume that
Then,
Therefore, the formula is valid for all positive integer values of n.
��k � 1��k � 2��k � 3�
3.
�k�k � 1��k � 2�
3�
3�k � 1��k � 2�3
� Sk � �k � 1��k � 2�
Sk�1 � 1�2� � 2�3� � 3�4� � . . . � k�k � 1� � �k � 1��k � 2�
Sk � 1�2� � 2�3� � 3�4� � . . . � k�k � 1� �k�k � 1��k � 2�
3.
n � 1, S1 � 2 �1�2��3�
3.
16. 1. When
2. Assume that
Then,
Therefore, the formula is valid for all positive integer values of n.
��k � 1�2�k � 2�2�2�k � 1�2 � 2�k � 1� � 1�
12.
��k � 1�2�k2 � 4k � 4��2k2 � 6k � 3�
12
��k � 1�2�2k4 � 14k3 � 35k2 � 36k � 12�
12
��k � 1�2�2k4 � 2k3 � k2 � 12�k3 � 3k2 � 3k � 1��
12
��k � 1�2�k2�2k2 � 2k � 1� � 12�k � 1�3�
12
�k2�k � 1�2�2k2 � 2k � 1�
12�
12�k � 1�5
12
Sk�1 � �k�1
i�1
i5 � �k
i�1
i5 � �k � 1�5
Sk � �k
i�1
i5 �k2�k � 1�2�2k2 � 2k � 1�
12.
n � 1, S1 ��1�2�1 � 1�2�2�1�2 � 2�1� � 1�
12� 1.
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Section 8.4 Mathematical Induction 729
18. 1. When
2. Assume that
Then,
Therefore, the formula is valid for all positive integer values of n.
�k � 1
2�k � 1� � 1.
��2k � 1��k � 1��2k � 1��2k � 3�
�2k2 � 3k � 1
�2k � 1��2k � 3�
�k�2k � 3� � 1
�2k � 1��2k � 3�
�k
2k � 1�
1
�2k � 1��2k � 3�
Sk�1 � Sk �1
�2�k � 1� � 1��2�k � 1� � 1�
Sk � �k
i�1
1
�2i � 1��2i � 1��
k
2k � 1.
S1 �1
�1��3� �1
2 � 1.n � 1,
19. 1. When
2. Assume
Thus,
Therefore, the formula is valid for all positive integer values of n.
�k � 1
�k � 1� � 1.
�k � 1k � 2
��k � 1�2
�k � 1��k � 2�
�k�k � 2� � 1
�k � 2��k � 2�
�k
k � 1�
1�k � 1��k � 2�
�1
1�2� �1
2�3� � . . . �1
k�k � 1� �1
�k � 1��k � 2�
Sk�1 � �k�1
i�1
1
i�i � 1�
Sk � �k
i�1
1
i�i � 1��
k
k � 1.
S1 �1
1�1 � 1� �12
n � 1,
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730 Chapter 8 Sequences, Series, and Probability
22.
�10�11��21��329�
30� 25,333
�10
n�1n4 �
10�10 � 1��2 � 10 � 1��3 � 102 � 3 � 10 � 1�30
23.
� 650 � 78 � 572
�12�12 � 1��2 � 12 � 1�
6�
12�12 � 1�2
�12
n�1�n2 � n� � �
12
n�1n2 � �
12
n�1n
24.
� 672,400 � 820 � 671,580
�402�40 � 1�2
4�
40�40 � 1�2
�40
n�1�n3 � n� � �
40
n�1n3 � �
40
n�1n 25. 1. When thus
2. Assume Then,since
Thus,
Therefore, by mathematical induction, the formula isvalid for all integers such that n ≥ 4.n
�k � 1�! > 2k�1.k � 1 > 2.�k � 1�! � k!�k � 1� > 2k�2�
k! > 2k, k > 4.
4! > 24.n � 4, 4! � 24 and 24 � 16,
20. 1. When
2. Assume
Thus,
Therefore, the formula is valid for all positive integer values of n.
��k � 1��k � 4�4�k � 2��k � 3�.
��k � 1�2�k � 4�
4�k � 1��k � 2��k � 3�
�k3 � 6k2 � 9k � 4
4�k � 1��k � 2��k � 3�
�k�k � 3��k � 3� � 4
4�k � 1��k � 2��k � 3�
�k�k � 3�
4�k � 1��k � 2� �1
�k � 1��k � 2��k � 3�
�1
1 � 2 � 3�
12 � 3 � 4
� . . . �1
k�k � 1��k � 2� �1
�k � 1��k � 2��k � 3�
Sk�1 � �k�1
i�1
1
i�i � 1��i � 2�
�k
i�1
1
i�i � 1��i � 2��
k�k � 3�4�k � 1��k � 2�
.
11�2��3� �
16
�1�4�
4�2��3�.n � 1,
21. �50
n�1n3 �
502�50 � 1�2
4� 1,625,625
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Section 8.4 Mathematical Induction 731
26. 1. When
2. Assume that
Then, Thus,
Therefore, 4
3n
> n.
43
k�1
> k � 1.4
3k�1
� 4
3k4
3 > k4
3 � k �k
3> k � 1 for k > 7.
4
3k
> k, k > 7.
4
37
� 7.4915 > 7.n � 7,
27. 1. When thus
2. Assume
Then,
Now we need to show that
This is true because
Therefore,
Therefore, by mathematical induction, the formula is valid for all integers n such that n ≥ 2.
1
�1�
1
�2�
1
�3� . . . �
1
�k�
1
�k � 1> �k � 1.
�k �1
�k � 1> �k � 1.
�k�k � 1� � 1
�k � 1>
k � 1
�k � 1
�k�k � 1� � 1 > k � 1
�k�k � 1� > k
�k �1
�k � 1> �k � 1, k > 2.
1
�1�
1
�2�
1
�3� . . . �
1
�k �
1
�k � 1> �k �
1
�k � 1.
1
�1�
1
�2�
1
�3� . . . �
1
�k> �k, k > 2.
1
�1�
1
�2 > �2.n � 2,
1
�1�
1
�2� 1.707 and �2 � 1.414,
28. 1. When
2. Assume that
Therefore, for all integers n ≥ 1.x
yn�1
< x
yn
x
yk�1
< x
yk ⇒ x
yx
yk�1
< x
yx
yk
⇒ x
yk�2
< x
yk�1
.
x
yk�1
< x
yk
x
y2
< x
y and �0 < x < y�.n � 1,
29. 1. When since
2. Assume
Then,
Therefore, by mathematical induction, the inequality is valid for all integers n ≥ 1.
� �k � 1�a.
� ka � ka2 ≥ ka � a �because a > 1�
�1 � a�k�1 � �1 � a�k�1 � a� ≥ ka�1 � a�
�1 � a�k ≥ ka.
1 > 0.n � 1, 1 � a ≥ a
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732 Chapter 8 Sequences, Series, and Probability
33. 1. When
2. Assume that
Then,
Thus, the formula is valid.
� x1�1x2
�1x3�1 . . . xk
�1xk�1�1.
� �x1x2x3 . . . xk��1xk�1�1
�x1x2x3 . . . xkxk�1��1 � ��x1x2x3
. . . xk�xk�1��1
�x1x2x3 . . . xk��1 � x1
�1x2�1x3
�1 . . . xk�1.
n � 1, �x1��1 � x1�1.
34. 1. When
2. Assume that
Thus, ln�x1x2 x3 . . . xn � � ln x1 � ln x2 � ln x3 � . . . � ln xn .
� ln x1 � ln x2 � ln x3 � . . . � ln xk � ln xk�1.
� ln�x1x2 x3 . . . xk� � ln xk�1
Then, ln�x1 x2 x3 . . . xk xk�1� � ln��x1x2x3 . . . xk�xk�1�
ln�x1 x2 x3 . . . xk� � ln x1 � ln x2 � ln x3 � . . . � ln xk .
ln x1 � ln x1.n � 1,
35. 1. When
2. Assume that
Then,
Hence, the formula holds.
� x�y1 � y2 � . . . � yk � yk�1�.
� x��y1 � y2 � . . . � yk� � yk�1�
xy1 � xy2 � . . . � xyk � xyk�1 � x�y1 � y2 � . . . � yk� � xyk�1
x�y1 � y2 � . . . � yk� � xy1 � xy2 � . . . � xyk.
n � 1, x�y1� � xy1.
30. 1. When
2. Assume that
First note that
Then,
Therefore for all integers n ≥ 1.3n > n2n
3k�1 � 3�3k� > 3�k2k� � �3k�2k ≥ 2�k � 1�2k � �k � 1�2k�1.
k ≥ 2 ⇒ 3k ≥ 2k � 2 � 2�k � 1�
3k > k2k, k ≥ 2.
31 > �1�21n � 1,
31. 1. When
2. Assume that
Then,
Thus, �ab�n � anbn.
� ak�1bk�1.
� akbkab
�ab�k�1 � �ab�k�ab�
�ab�k � akbk.
n � 1, �ab�1 � a1b1 � ab. 32. 1. When
2. Assume that
Thus, a
bn
�an
bn.
�ak�1
bk�1.�
ak
bk �a
bThen, a
bk�1
� a
bka
b
a
bk
�ak
bk.
a
b1
�a1
b1.n � 1,
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Section 8.4 Mathematical Induction 733
36. 1. When and are complex conjugates by definition.
2. Assume that and are complex conjugates.
That is, if then
Then,
This implies that and are complex conjugates. Therefore, and are complex conjugates for n ≥ 1.�a � bi�n�a � bi�n
�a � bi�k�1�a � bi�k�1
� �ac � bd � � i�bc � ad �.
and �a � bi�k�1 � �a � bi�k �a � bi� � �c � di��a � bi�
� �ac � bd � � i�bc � ad �
�a � bi�k�1 � �a � bi�k�a � bi� � �c � di��a � bi�
�a � bi�k � c � di .�a � bi�k � c � di,
�a � bi�k�a � bi�k
a � bia � bin � 1,
37. 1. When is a factor.
2. Assume that 3 is a factor of
Then,
Since 3 is a factor of by our assumption, and 3 is a factor of then 3 is a factor of the whole sum.
Thus, 3 is a factor of for every positive integer n.�n3 � 3n2 � 2n�
3�k2 � 3k � 2��k3 � 3k2 � 2k�
� �k3 � 3k2 � 2k� � 3�k2 � 3k � 2�.
� �k3 � 3k2 � 2k� � �3k2 � 9k � 6�
� k3 � 3k2 � 3k � 1 � 3k2 � 6k � 3 � 2k � 2��k � 1�3 � 3�k � 1�2 � 2�k � 1��
�k3 � 3k2 � 2k�.
n � 1, �13 � 3�1�2 � 2�1�� � 6 and 3
38. 1. When 3 is a factor of
2. Assume that 3 is a factor of
Then,
Because 3 is a factor of both terms, 3 is a factor of
Therefore, 3 is a factor of for all n > 0.n3 � 5n � 6
�k � 1�3 � 5�k � 1� � 6.
� �k3 � 5k � 6� � 3�k2 � k � 2�.
� �k3 � 5k � 6� � �3k2 � 3k � 6�
� k3 � 3k2 � 8k � 12
�k � 1�3 � 5�k � 1� � 6 � k3 � 3k2 � 3k � 1 � 5k � 11
k3 � 5k � 6.
�13 � 5�1� � 6� � 12.n � 1,
39. 1. When and 3 is a factor.
2. Assume that 3 is a factor of Then,
Since 3 is a factor of by our assumption, and 3 is a factor of then 3 is a factor of the whole sum.
Thus, 3 is a factor of for every positive integer n.n3 � n � 3
3�k2 � k�,k3 � k � 3
� �k3 � k � 3� � 3�k2 � k�.
� �k3 � k � 3� � 3k2 � 3k
� k3 � 3k2 � 2k � 3
��k � 1�3 � �k � 1� � 3� � k3 � 3k2 � 3k � 1 � k � 1 � 3
k3 � k � 3.
�13 � 1 � 3� � 3,n � 1,
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734 Chapter 8 Sequences, Series, and Probability
41. 1. When and 3 is a factor.
2. Assume that 3 is a factor of
Then,
Since 3 is a factor of by our assump-tion, and 3 is a factor of then 3 is afactor of the whole sum.
Thus, 3 is a factor of for every positiveinteger n.
22n�1 � 1
3 � 22k�1,22k�1 � 1
� �22k�1 � 1� � 3 � 22k�1.
� �3 � 1�22k�1 � 1
� 4 � 22k�1 � 1
22�k�1��1 � 1 � 22k�3 � 1
22k�1 � 1.
22�1 � 1 � 9,n � 1, 42. 1. When and 5 is a factor.
2. Assume that 5 is a factor of
Then,
Since 5 is a factor of by our assump-tion, and 5 is a factor of then 5 is afactor of the whole sum.
Thus, 5 is a factor of for every positiveinteger n.
24n�2 � 1
15 � 24k�2,24k�2 � 1
� �24k�2 � 1� � 15 � 24k�2.
� �15 � 1�24k�2 � 1
� 16 � 24k�2 � 1
24�k�1��2 � 1 � 24k�2 � 1
24k�2 � 1.
24�1��2 � 1 � 5,n � 1,
43.
First differences:
Second differences:
Since the first differences are equal, the sequence has a linear model.
an:
a5 � a4 � 3 � 9 � 3 � 12
a4 � a3 � 3 � 6 � 3 � 9
a3 � a2 � 3 � 3 � 3 � 6
a2 � a1 � 3 � 0 � 3 � 3
a1 � 0
a1 � 0, an � an�1 � 3
0 3 6 9 12
3 3 3 3
0 0 0
44.
2 0 3 1 4
First differences: 3 3
Second differences: 5 5
Since neither the first differences nor the seconddifferences are equal, the sequence does not have a linear or quadratic model.
�5
�2�2
an:
a5 � n � a5 � 5 � 1 � 4
a4 � n � a3 � 4 � 3 � 1
a3 � n � a2 � 3 � 0 � 3
a2 � n � a1 � 2 � 2 � 0
a1 � 2
a1 � 2, an � n � an�1
40. 1. When and 2 is a factor.
2. Assume that 2 is a factor of
Then,
Since 2 is a factor of each term, it is a factor of the sum.
Thus, 2 is a factor of for each positive integer n.n4 � n � 4
� �k4 � k � 4� � �4k3 � 6k2 � 4k�.
� k4 � 4k3 � 6k2 � 3k�4
��k � 1�4 � �k � 1� � 4� � �k4 � 4k3 � 6k2 � 4k � 1� � k � 1 � 4
k4 � k � 4.
�14 � 1 � 4� � 4,n � 1,
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Section 8.4 Mathematical Induction 735
45.
First differences:
Second differences:
Since the second differences are all the same,the sequence has a quadratic model.
an:
a5 � a4 � 5 � �6 � 5 � �11
a4 � a3 � 4 � �2 � 4 � �6
a3 � a2 � 3 � 1 � 3 � �2
a2 � a1 � 2 � 3 � 2 � 1
a1 � 3
a1 � 3, an � an�1 � n
3 1
�1�1�1
�5�4�3�2
�11�6�2
46.
6 24
First differences: 36
Second differences: 54
Since neither the first nor the second differences are equal, the sequence does not have a linear orquadratic model.
�108�27
�72�189
�48�12�3an:
a6 � �2a5 � �2�24� � �48
a5 � �2a4 � �2��12� � 24
a4 � �2a3 � �2�6� � �12
a3 � �2a2 � �2��3� � 6
a2 � �3
a2 � �3, an � �2an�1
47.
First differences:
Second differences:
Since the second differences are equal, the sequence has a quadratic model.
an:
a4 � a3 � 4 � 6 � 4 � 10
a3 � a2 � 3 � 3 � 3 � 6
a2 � a1 � 2 � 1 � 2 � 3
a1 � a0 � 1 � 0 � 1 � 1
a0 � 0
a0 � 0, an � an�1 � n
49.
First differences:
Second differences:
Since the first differences are equal, the sequencehas a linear model.
an:
a5 � a4 � 2 � 8 � 2 � 10
a4 � a3 � 2 � 6 � 2 � 8
a3 � a2 � 2 � 4 � 2 � 6
a2 � a1 � 2 � 2 � 2 � 4
a1 � 2
a1 � 2, an � an�1 � 2
2 4 6 8 10
2 2 2 2
0 0 0
0 1 3 6 10
1 2 3 4
1 1 1
48.
2 4 16 256 65,536
First differences: 2 12 240 65,280
Second differences: 10 228 65,040
Since neither the first differences nor the seconddifferences are equal, the sequence does not have a linear or quadratic model.
an:
a4 � a32 � 2562 � 65,536
a3 � a22 � 162 � 256
a2 � a12 � 42 � 16
a1 � a02 � 22 � 4
a0 � 2
a0 � 2, an � �an�1�2
50.
0 4 10 18 28
First differences: 4 6 8 10
Second differences: 2 2 2
Since the second differences are equal, the sequencehas a quadratic model.
an:
a5 � a4 � 2�5� � 18 � 10 � 28
a4 � a3 � 2�4� � 10 � 8 � 18
a3 � a2 � 2�3� � 4 � 6 � 10
a2 � a1 � 2�2� � 0 � 4 � 4
a1 � 0
a1 � 0, an � an�1 � 2n
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736 Chapter 8 Sequences, Series, and Probability
53.
Let Then:
By elimination:
Thus, an �12n2 � n � 3.
a � 12 ⇒ b � 1
2a � 1
4a � b � 3
�2a � b � �2
4a � b � 3
16a � 4b � 12
16a � 4b � c � 9a4 � a�4�2 � b�4� � c � �9 ⇒
2a � b � 2
4a � 2b � 4
4a � 2b � c � 1 a2 � a�2�2 � b�2� � c � �1 ⇒
a0 � a�0�2 � b�0� � c � �3 ⇒ c � �3
an � an2 � bn � c.
a0 � �3, a2 � 1, a4 � 9
54.
Let Thus:
By elimination:
Thus, an �74 n2 � 5n � 3.
�4a12a8a
�
�
2b2b
a
�
�
�
�
31114
74 ⇒ b � �5
a0
a2
a6
�
�
�
a�0�2
a�2�2
a�6�2
�
�
�
b�0�b�2�
b�6�
�
�
�
cc
c
�
�
�
30
36
⇒ ⇒
⇒
4a4a
36a36a12a
�
�
�
�
�
2b2b6b6b2b
�
�
cc
c
�
�
�
�
�
�
30
�3363311
an � an2 � bn � c.
a0 � 3, a2 � 0, a6 � 36
51.
Let
Solving the system,an � n2 � 3n � 5, n ≥ 1.
a � 1, b � �3, c � 5,
a3 � a�3�2 � b�3� � c � 5 ⇒ 9a � 3b � c � 5
a2 � a�2�2 � b�2� � c � 3 ⇒ 4a � 2b � c � 3
a1 � a�1�2 � b�1� � c � 3 ⇒ a � b � c � 3
an � an2 � bn � c.
a1 � 3, a2 � 3, a3 � 5 52.
Let
Solving the system,an � n2 � 4n � 10, n ≥ 1.
a � 1, b � �4, c � 10,
a3 � a�3�2 � b�3� � c � 7 ⇒ 9a � 3b � c � 7
a2 � a�2�2 � b�2� � c � 6 ⇒ 4a � 2b � c � 6
a1 � a�1�2 � b�1� � c � 7 ⇒ a � b � c � 7
an � an2 � bn � c.
a1 � 7, a2 � 6, a3 � 7
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Section 8.4 Mathematical Induction 737
56. (a) One ring one move
Two rings three moves
Three rings seven moves
(b) Four rings: 7 moves to move 3
1 move for fourth ring
7 moves to bring back 3
Total: 15 moves
→
→
→ (c) If n rings, let be the number of moves. Then,
(d) 1. For one ring,
2. Assume that For rings, it takesmoves to move n rings, one to move the last ring, andmore to move the n rings back.
Total: hn�1 � 2�hn� � 1
hn
hn
n � 1hn � 2hn�1 � 1.
h1 � 1.
hn � 2hn�1 � 1 � 2n � 1 moves.
h1 � 1
hn
57. False. might not even bedefined.
P1 59. False. It has second differences.
n � 258. False. See the Study Tip onpage 550.
60. (a) If is true and implies then is true for integers
(b) If are all true, then is true for integers
(c) If and are all true, but the truth of does not imply that is true, then youmay only conclude that and are true.
(d) If is true and implies then is true for any positive integer n.P2nP2k�2,P2kP2
P3P2,P1,Pk�1PkP3P2,P1,
1 ≤ n ≤ 50.PnP50. . .,P3,P2,P1,
n ≥ 3.PnPk�1,PkP3
61. �2x2 � 1�2 � 4x4 � 4x2 � 1
63. �5 � 4x�3 � �64x3 � 240x2 � 300x � 125
65. � 9�3i � 2�3i � 7�3i 3��27 � ��12 � 3�3 � 3 � ��3� � �2 � 2��3�
62. �2x � y�2 � 4x2 � 4xy � y2
64. �2x � 4y�3 � 8x3 � 48x2y � 96xy2 � 64y3
66. � �3 � 6 3�23 �125 � 4 3��8 � 2 3��54 � 5 � 4��2� � 6 3�2
55. (a) 3 sides
Koch snowflake: sides
To prove this, use mathematical induction.
1. For the number of sides is
2. Assume that the number of sides of the Kochsnowflake is When the Kochsnowflake is created, each side is replaced with 4 sides. That is, the number of sides is increasedby a factor of 4:
Number sides
Hence, the formula is valid for all positive integers n.
� 4�3 � 4k�1� � 3 � 4k.
�k � 1�st3 � 4k�1.kth
3 � 41�1 � 3.n � 1,
3�4�n�1nth
3 � 42 � 48 sidesn � 3:
3 � 4 � 12 sidesn � 2:
n � 1: (b)
(c) For the nth Koch snowflake, the length of a single side is and the number of sides is Hence, the perimeter is
13
n�1
3 � 4n�1 � 343
n�1
.
3 � 4n�1.�1�3�n�1,
An ��34 �1 � �
n
k�2
13
49
k�2�, n > 1
n � 4: A4 ��34 �1 �
13
�13
49 �
13
49
2�
n � 3: A3 ��34 �1 �
13
�13
49�
n � 2: A2 ��34 �1 �
13�
n � 1: A1 ��34
�1�2 ��34
67.
� 40�1 � 3�2 � � 40 � 40 3��2
10� 3�64 � 2 3��16 � � 10�4 � 22 3��2 � 68.
� 25 � 9 � 30i � 16 � 30i
��5 � ��9�2� ��5 � 3i�2
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Section 8.5 The Binomial Theorem
738 Chapter 8 Sequences, Series, and Probability
■ You should be able to use the Binomial Theorem
where to expand
■ You should be able to use Pascal’s Triangle.
�x � y�n.nCr �n!
�n � r�!r!,
�x � y�n � xn � nxn�1y �n�n � 1�
2!xn�2y2 � . . . � nCr xn�ryr � . . . � yn
1. 7C5 �7!
2!5!�
7 � 6 � 5!
2 � 5!�
42
2� 21
3. �12
0 � � 12C0 �12!
0!12!� 1
9. �100
98 � � 100C98 �100!
98!2!�
100 � 99
2 � 1� 4950
5. 20C15 �20!
15!5!�
20 � 19 � 18 � 17 � 16
5 � 4 � 3 � 2 � 1� 15,504
7. 14C1 �14!
13!1!�
14 � 13!13!
� 14
2. 9C6 �9!
6!3!�
9 � 8 � 7 � 6!6! 3 � 2
�9 � 8 � 7
6� 84
4. �2020� � 20C20 �
20!20!0!
� 1
6. 12C3 �12!9!3!
�12 � 11 � 10 � 9!
9! 3 � 2� 220
8. 18C17 �18!
17!1!�
18 � 17!17!
� 18
10. �107 � �
10!7!3!
�10 � 9 � 8 � 7!
7! 3 � 2� 120
13. 100C98 � 495012. 34C4 � 46,37611. 41C36 � 749,398
16. 1000C2 � 499,50015. 250C2 � 31,12514. 500C498 � 124,750
17.
� x4 � 8x3 � 24x2 � 32x � 16
�x � 2�4 � 4C0x4 � 4C1x
3�2� � 4C2x2�2�2 � 4C3x�2�3 � 4C4�2�4
19.
� a3 � 3a2�3� � 3a�3�2 � �3�3 � a3 � 9a2 � 27a � 27
�a � 3�3 � 3C0a3 � 3C1a
2�3� � 3C2a�3�2 � 3C3�3�3
18.
� x6 � 6x5 � 15x 4 � 20x3 � 15x2 � 6x � 1
�x � 1�6 � 6C0 x6 � 6C1x5�1� � 6C2x
4�1�2 � 6C3x3�1�3 � 6C4x
2�1�4 � 6C5x�1�5 � 6C6�1�6
Vocabulary Check
1. binomial coefficients 2. Binomial Theorem, Pascal’s Triangle
3. or 4. expanding, binomial�nr�nCr
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Section 8.5 The Binomial Theorem 739
20. � a4 � 8a3 � 24a2 � 32a � 16�a � 2�4 � 4C0a4 � 4C1a
3�2� � 4C2a2�2�2 � 4C3a�2�3 � 4C4�2�4
21.
� y4 � 8y3 � 24y2 � 32y � 16
� y4 � 4y3�2� � 6y2�4� � 4y�8� � 16
�y � 2�4 � 4C0y4 � 4C1y
3�2� � 4C2y2�2�2 � 4C3y�2�3 � 4C4�2�4
22.
� y5 � 10y4 � 40y3 � 80y2 � 80y � 32
� y � 2�5 � 5C0 y5 � 5C1y4�2� � 5C2 y3�2�2 � 5C3y
2�2�3 � 5C4y�2�4 � 5C5�2�5
24.
� x6 � 6x5y � 15x4y2 � 20x3y3 � 15x2y4 � 6xy5 � y6
�x � y�6 � 6C0 x6 � 6C1x5y � 6C2x
4y2 � 6C3x3y3 � 6C4x
2y4 � 6C5xy5 � 6C6 y6
26. �4x � 3y�4 � 256x4 � 768x3y � 864x2y2 � 432xy3 � 81y4
28.
� 32x5 � 80x4y � 80x3y2 � 40x2y3 � 10xy4 � y5
� 32x5 � 5�16x4�y � 10�8x3�y2 � 10�4x2�y3 � 5�2x�y4 � y5
�2x � y�5 � 5C0�2x�5 � 5C1�2x�4y � 5C2�2x�3y2 � 5C3�2x�2y3 � 5C4�2x�y4 � 5C5y5
23.
� x5 � 5x4y � 10x3y2 � 10x2y3 � 5xy4 � y5
�x � y�5 � 5C0x5 � 5C1x
4y � 5C2x3y2 � 5C3x
2y3 � 5C4xy4 � 5C5y5
25.
� 729r6 � 2916r5s � 4860r4s2 � 4320r3s3 � 2160r2s4 � 576rs5 � 64s6
� 6C4�3r�2�2s�4 � 6C5�3r��2s�5 � 6C6�2s�6�3r � 2s�6 � 6C0�3r�6 � 6C1�3r�5�2s� � 6C2�3r�4�2s�2 � 6C3�3r�3�2s�3
27.
� x5 � 5x4y � 10x3y2 � 10x2y3 � 5xy4 � y5
�x � y�5 � 5C0x5 � 5C1x
4y � 5C2x3y2 � 5C3x
2y3 � 5C4xy4 � 5C5y5
29.
� 1 � 12x � 48x2 � 64x3
� 1 � 3�4x� � 3�4x�2 � �4x�3
�1 � 4x�3 � 3C013 � 3C11
2�4x� � 3C21�4x�2 � 3C3�4x�3
30. �5 � 2y�3 � 125 � 150y � 60y2 � 8y3
31.
� x8 � 8x6 � 24x4 � 32x2 � 16
�x2 � 2�4 � 4C0�x2�4 � 4C1�x2�3�2� � 4C2�x2�222 � 4C3�x2�23 � 4C4�2�4
32.
� �y6 � 9y4 � 27y2 � 27
� 27 � 27y2 � 9y4 � y6
�3 � y2�3 � 3C0�3�3 � 3C1�3�2y2 � 3C2�3��y2�2 � 3C3�y2�3
33.
� x10 � 25x8 � 250x6 � 1250x4 � 3125x2 � 3125
�x2 � 5�5 � 5C0�x2�5 � 5C1�x2�4�5� � 5C2�x2�3�52� � 5C3�x2�2�53� � 5C4�x2��54� � 5C5�55�
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740 Chapter 8 Sequences, Series, and Probability
34.
� y12 � 6y10 � 15y8 � 20y6 � 15y4 � 6y2 � 1
�y2 � 1�6 � 6C0�y2�6 � 6C1�y2�5 � 6C2�y2�4 � 6C3�y2�3 � 6C4�y2�2 � 6C5�y2� � 6C6
36.
� x12 � 6x10y2 � 15x8y4 � 20x6y6 � 15x4y8 � 6x2y10 � y12
� 6C5�x2�� y2�5 � 6C6� y2�6 �x2 � y2�6 � 6C0�x2�6 � 6C1�x2�5� y2� � 6C2�x2�4� y2�2 � 6C3�x2�3� y2�3 � 6C4�x2�2� y2�4
35.
� x8 � 4x6y2 � 6x4y4 � 4x2y6 � y8
�x2 � y2�4 � 4C0�x2�4 � 4C1�x2�3�y2� � 4C2�x2�2�y2�2 � 4C3�x2��y2�3 � 4C4�y2�4
37.
� x18 � 6x15y � 15x12y2 � 20x9y3 � 15x6y4 � 6x3y5 � y6
� 6C4�x3�2y4 � 6C5�x3�y5 � 6C6 y6 �x3 � y�6 � 6C0�x3�6 � 6C1�x3�5 y � 6C2�x3�4y2 � 6C3�x3�3 y3
38.
� 32x15 � 80x12y � 80x9y2 � 40x6y3 � 10x3y4 � y5
� 5C3�2x3�2y3 � 5C4�2x3�y4 � 5C5y5�2x3 � y�5 � 5C0�2x3�5 � 5C1�2x3�4y � 5C2�2x3�3y2
40.
�1
x6�
12y
x5�
60y2
x4�
160y3
x3�
240y4
x2�
192y5
x� 64y6
� 6�32��1
x�y5 � 1�64�y6 � 1�1
x�6
� 6�2��1
x�5y � 15�4��1
x�4y2 � 20�8��1
x�3y3 � 15�16��1
x�2y4
� 6C4�1
x�2
�2y�4 � 6C5�1
x��2y�5 � 6C6�2y�6�1
x� 2y�6
� 6C0�1
x�6
� 6C1�1
x�5�2y� � 6C2�1
x�4�2y�2 � 6C3�1
x�3�2y�3
39.
�1
x5�
5y
x4�
10y2
x3�
10y3
x2�
5y4
x� y5
�1
x� y�
5
� 5C0�1
x�5
� 5C1�1
x�4
y � 5C2�1
x�3
y2 � 5C3�1
x�2
y3 � 5C4�1
x�y4 � 5C5y5
41.
�16x4 �
32x3 y �
24x2 y2 �
8x
y3 � y4
�2x
� y�4� 4C0�2
x�4
� 4C1�2x�
3y � 4C2�2
x�2y2 � 4C3�2
x�y3 � 4C4y4
42.
�32x5 �
240x4 y �
720x3 y2 �
1080x2 y3 �
810x
y4 � 243y5
� 5C3�2x�
2�3y�3 � 5C4�2
x��3y�4 � 5C5�3y�5�2x
� 3y�5� 5C0�2
x�5
� 5C1�2x�
4�3y� � 5C2�2
x�3�3y�2
43.
� �512x4 � 576x3 � 240x2 � 44x � 3
�4x � 1�3 � 2�4x � 1�4 � �64x3 � 48x2 � 12x � 1� � 2�256x4 � 256x3 � 96x2 � 16x � 1�
44.
� x5 � 11x4 � 42x3 � 54x2 � 27x � 81
�x � 3�5 � 4�x � 3�4 � �x5 � 15x4 � 90x3 � 270x2 � 405x � 243� � 4�x4 � 12x3 � 54x2 � 108x � 81�
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Section 8.5 The Binomial Theorem 741
45.
� 2x4 � 24x3 � 113x2 � 246x � 207
� 2�x4 � 12x3 � 54x2 � 108x � 81� � 5�x2 � 6x � 9�
� 5�x2 � 2�x��3� � 32�2�x � 3�4 � 5�x � 3�2 � 2�x4 � 4�x3��3� � 6�x2��32� � 4�x��33� � 34�
46.
� 3x5 � 15x4 � 34x3 � 42x2 � 27x � 7
3�x � 1�5 � 4�x � 1�3 � �3x5 � 15x4 � 30x3 � 30x2 � 15x � 3� � �4x3 � 12x2 � 12x � 4�
47.
� �4x6 � 24x5 � 60x4 � 83x3 � 42x2 � 60x � 20
�3�x � 2�3 � 4�x � 1�6 � ��3x3 � 18x2 � 36x � 24� � �4x6 � 24x5 � 60x4 � 80x3 � 60x2 � 24x � 4�
48.
� 5x5 � 50x4 � 200x3 � 398x2 � 404x � 158
5�x � 2�5 � 2�x � 1�2 � �5x5 � 50x4 � 200x3 � 400x2 � 400x � 160� � �2x2 � 4x � 2�
49.
10C3x10�3�8�3 � 120x7�512� � 61,440x7
�x � 8�10, n � 4
51.
5C2x5�2��6y�2 � 10x3�36�y2 � 360x3y2
�x � 6y�5, n � 3
50.
6C6 x0��5�6 � 15,625
�x � 5�6, n � 7
52.
7C3 x7�3��10z�3 � �35,000 x4z3
�x � 10z�7, n � 4
53.
� 1,259,712x2y7
9C7�4x�9�7�3y�7 � 36�16�x2�37�y7
�4x � 3y�9, n � 8
55.
� 32,476,950,000x4y8
12C8�10x�12�8��3y�8 � 495�104��38�x4y8
�10x � 3y�12, n � 9
54.
5C4�5a�5�4�6b�4 � 32,400 a � b4
�5a � 6b�5, n � 5
56.
15C7�7x�15�7�2y�7 � 4.7 � 1012 x8y7
�7x � 2y�15, n � 8
57. The term involving in the expansion of is The coefficient is 3,247,695.
12C8x4�3�8 � 495x4�3�8 � 3,247,695x4.
�x � 3�12x4 58.
a � 12,976,128
12C7 x5�4�7 � 12,976,128x5
�x � 4�12, ax5
59. The term involving in the expansion of is
The coefficient is 180.
10C2x8��2y�2 �
10!
2!8!� 4x8y2 � 180x8y2.
�x � 2y�10x8 y2 60. The term involving in the expansion of
is
The coefficient is 720.
10C8�4x�2��y�8 �10!
�10 � 8�!8!� 16x2y8 � 720x2y8.
�4x � y�10x2y8
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742 Chapter 8 Sequences, Series, and Probability
63. The coefficient of in the expansionof is 10C6 � 210.�x2 � y�10
� �x2�4y6x8 y6 64. The term involving in the expansion of
is
The coefficient is �220.
12C9�z2�3��1�9 �12
�12 � 9�!9!z6��1� � �220z6.
�z2 � 1�12z6
65. 5th entry of 7th row: 7C5 � 21 66. 3rd entry of 6th row: 6C3 � 20
67. 5th entry of 6th row: 6C5 � 6 68. 2nd entry of 5th row: 5C2 � 10
69. 4th row of Pascal’s Triangle: 1 4 6 4 1
� 81t4 � 216t3v � 216t2v2 � 96tv3 � 16v4
�3t � 2v�4 � 1�3t�4 � 4�3t�3�2v� � 6�3t�2�2v�2 � 4�3t��2v�3 � 1�2v�4
70. 4th row of Pascal’s Triangle: 1 4 6 4 1
� 625v4 � 1000v3z � 600v2z2 � 160vz3 � 16z4
�5v � 2z�4 � 1�5v�4 � 4�5v�3�2z� � 6�5v�2�2z�2 � 4�5v��2z�3 � 1�2z�4
71. 5th row of Pascal’s Triangle: 1 5 10 10 5 1
� 32x5 � 240x4y � 720x3y2 � 1080x2y3 � 810xy4 � 243y5
�2x � 3y�5 � 1�2x�5 � 5�2x�4�3y� � 10�2x�3�3y�2 � 10�2x�2�3y�3 � 5�2x��3y�4 � �3y�5
72. 5th row of Pascal’s Triangle: 1 5 10 10 5 1
� 3125y5 � 6250y4 � 5000y3 � 2000y2 � 400y � 32
�5y � 2�5 � 1�5y�5 � 5�5y�42 � 10�5y�322 � 10�5y�223 � 5�5y�24 � 25
73.
� x2 � 20x3�2 � 150x � 500x1�2 � 625
� x2 � 20xx � 150x � 500x � 625
�x � 5�4 � �x�4� 4�x�3�5� � 6�x�2�5�2 � 4�x��53� � 54
74.
� 64tt � 48t � 12t � 1 � 64t3�2 � 48t � 12t1�2 � 1
�4t � 1�3� �4t �3
� 3�4t �2 ��1� � 3�4t���1�2 � ��1�3
75.
� x2 � 3x4�3y1�3 � 3x2�3y2�3 � y
�x2�3 � y1�3�3 � �x2�3�3 � 3�x2�3�2 � y1�3� � 3�x2�3� � y1�3�2 � � y1�3�3
76. �u3�5 � v1�5�5 � u3 � 5u12�5v1�5 � 10u9�5v2�5 � 10u6�5v3�5 � 5u3�5v4�5 � v
61. The term involving in is
The coefficient is �489,888.
� �489,888x6y3.
9C3�3x�6��2y�3 � 84�3�6��2�3x6y3
�3x � 2y�9x6y3 62. The term involving in the expansion ofis
a � 90,720
� 90,720x4y4.
8C4�2x�4��3y�4 � 70�24���3�4x4y4
�2x � 3y�8x4y4
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Section 8.5 The Binomial Theorem 743
77.
� 3x2 � 3xh � h2, h � 0
�h�3x2 � 3xh � h2�
h
�x3 � 3x2h � 3xh2 � h3 � x3
h
f�x � h� � f�x�h
��x � h�3 � x3
h
78.
� 4x3 � 6x2h � 4xh2 � h3, h � 0
�h�4x3 � 6x2h � 4xh2 � h3�
h
�x 4 � 4x3h � 6x2h2 � 4xh3 � h4 � x 4
h
f �x � h� � f �x�h
��x � h�4 � x 4
h
79.
� 6x5 � 15x4h � 20x3h2 � 15x2h3 � 6xh4 � h5, h � 0
�h�6x5 � 15x4h � 20x3h2 � 15x2h3 � 6xh4 � h5�
h
��x6 � 6x5h � 15x4h2 � 20x3h3 � 15x2h4 � 6xh5 � h6� � x6
h
f�x � h� � f�x�h
��x � h�6 � x6
h
80.
� 8x7 � 28x6h � 56x5h2 � 70x4h3 � 56x3h4 � 28x2h5 � 8xh6 � h7, h � 0
�h�8x7 � 28x6h � 56x5h2 � 70x4h3 � 56x3h4 � 28x2h5 � 8xh6 � h7�
h
��x8 � 8x7h � 28x6h2 � 56x5h3 � 70x4h4 � 56x3h5 � 28x2h6 � 8xh7 � h8� � x8
h
f�x � h� � f�x�h
��x � h�8 � x8
h
81.
�1
x � h � x, h � 0
��x � h� � x
h�x � h � x�
�x � h � x
h�x � h � xx � h � x
f�x � h� � f�x�h
�x � h � x
h82.
� �1
x�x � h�, h � 0
�
�h
x�x � h�h
�
x � �x � h�x�x � h�
h
f �x � h� � f �x�h
�
1
x � h�
1
x
h
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744 Chapter 8 Sequences, Series, and Probability
83.
� �4
� 1 � 4i � 6 � 4i � 1
�1 � i�4 � 4C014 � 4C1�1�3i � 4C2�1�2i2 � 4C31 � i3 � 4C4i
4
84.
� 404 � 1121i
� 1024 � 1280i � 640 � 160i � 20 � i
�4 � i�5 � 1024 � 1280i � 640i2 � 160i3 � 20i4 � i5
85.
� 161 � 240i
� 256 � 256i � 96 � 16i � 1
�4 � i�4 � 4C0�4�4 � 4C1�43�i � 4C2�42��i2� � 4C3�4��i3� � 4C4i4
86.
� �38 � 41i
� 32 � 80i � 80 � 40i � 10 � i
�2 � i�5 � 25 � 5�24�i � 10�23��i2� � 10�22��i3� � 5�2��i4� � i5
87.
� 2035 � 828i
� 64 � 576i � 2160 � 4320i � 4860 � 2916i � 729
�2 � 3i�6 � 6C026 � 6C12
5�3i� � 6C224�3i�2 � 6C32
3�3i�3 � 6C422�3i�4 � 6C52�3i�5 � 6C6�3i�6
88.
� �2035 � 828i
� 729 � 2916i � 4860 � 4320i � 2160 � 576i � 64
�3 � 2i�6 � 36 � 6�35��2i� � 15�34��2i�2 � 20�33��2i�3 � 15�32��2i�4 � 6�3��2i�5 � �2i�6
89.
� �115 � 236i
� 125 � 300i � 240 � 64i
� 53 � 3�52��4i� � 3�5��4i�2 � �4i�3
�5 � �16�3 � �5 � 4i�3 90.
� �10 � 198i
� 125 � 225i � 135 � 27i
� 53 � 3 � 52�3i� � 3 � 5�3i�2 � �3i�3
�5 � �9 �3 � �5 � 3i�3
91.
� �23 � 2083i
� 256 � 2563i � 288 � 483i � 9
�4 � 3i�4 � 44 � 4�43��3i� � 6�42��3i�2 � 4�4��3i�3 � �3�4
92.
� 184 � 4403i
� 625 � 5003i � 450 � 603i � 9
�5 � 3i�4 � 54 � 4 � 53�3i� � 6 � 52�3i�2 � 4 � 5�3i�3 � �3i�4
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Section 8.5 The Binomial Theorem 745
93.
� 1
�1
8��1 � 33i � 9 � 33i�
�1
8���1�3 � 3��1�2�3i� � 3��1��3i�2
� �3i�3�
��1
2�
3
2i�
3
�1
8��1 � 3i�3
94.
�18
�33
8i �
98
�33
8i � �1
�12
�32
i�3
� �12�
3
� 3�12�
2�32
i� � 3�12��3
2i�2
� �32
i�3
95.
� �18
� 164
�34�
316�� � �3
16 34
�3364
�i
�14
�34
i�3
� �14�
3
� 3�14�
2�34
i� � 3�14���3
4i�2
� �34
i�3
96.
�127
�39
i �13
�39
i � �827
�13
�33
i�3
� �13�
3
� 3�13�
2�3i3 � � 3�1
3��33
i�2
� �33
i�3
97.
� 1 � 0.16 � 0.0112 � 0.000448 � . . . � 1.172
� � 28�0.02�6 � 8�0.02�7 � �0.02�8
�1.02 �8 � �1 � 0.02�8 � 1 � 8�0.02� � 28�0.02�2 � 56�0.02�3 � 70�0.02�4 � 56�0.02�5
98.
� 1049.890
� 1024 � 25.6 � 0.288 � 0.00192 � 0.0000084 � . . .
� 10�2��0.005�9 � �0.005�10
� 252�2�5�0.005�5 � 210�2�4�0.005�6 � 120�2�3�0.005�7 � 45�2�2�0.005�8
�2.005�10 � �2 � 0.005�10 � 210 � 10�2�9�0.005� � 45�2�8�0.005�2 � 120�2�7�0.005�3 � 210�2�6�0.005�4
99.
� 510,568.785
� 220�3�3�0.01�9 � 66�3�2�0.01�10 � 12�3��0.01�11 � �0.01�12
� 792�3�7�0.01�5 � 924�3�6�0.01�6 � 792�3�5�0.01�7 � 495�3�4�0.01�8
� 312 � 12�3�11�0.01� � 66�3�10�0.01�2 � 220�3�9�0.01�3 � 495�3�8�0.01�4
�2.99�12 � �3 � 0.01�12
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746 Chapter 8 Sequences, Series, and Probability
100.
� 467.721
� 512 � 46.08 � 1.8432 � 0.043008 � 0.00064512
� 126�2�4�0.02�5 � 84�2�3�0.02�6 � 36�2�2�0.02�7 � 9�2��0.02�8 � �0.02�9
�1.98�9 � �2 � 0.02�9 � 29 � 9�2�8�0.02� � 36�2�7�0.02�2 � 84�2�6�0.02�3 � 126�2�5�0.02�4
101.
g is shifted three units to the left.
−6
−10 8
6
fg
� x3 � 9x2 � 23x � 15
� x3 � 9x2 � 27x � 27 � 4x � 12
� �x � 3�3 � 4�x � 3�
g�x� � f �x � 3�
f�x� � x3 � 4x
103.
Since is the expansion of they have thesame graph.
f�x�,p�x�
p�x� � 1 � 3x � 3x2 � x3
h�x� � 1 � 3x � 3x2
g�x� � 1 � 3x
−3
−6 6
5
h
f = p
gf�x� � �1 � x�3
102.
is shifted five units to the right of
−6
−6 12
6
gf
f.g
� �x4 � 20x3 � 146x2 � 460x � 526
� 4�x2 � 10x � 25� � 1
� ��x4 � 20x3 � 150x2 � 500x � 625�
� ��x � 5�4 � 4�x � 5�2 � 1
g�x� � f �x � 5�
f �x� � �x 4 � 4x2 � 1
104.
is the expansion of
−3
−4 8
5
h
f = pg
f �x�.p�x�
p�x� � 1 � 2x �32x2 �
12x3 �
116x4 � f�x�
105. 7C4�12�4�1
2�3� 35� 1
16��18� � 0.273
107. 8C4�13�4�2
3�4� 70� 1
81��1681� � 0.171
106. 10C3�14�3�3
4�7� 120� 1
64�� 218716,384� � 0.2503
108. 8C4�12�4�1
2�4� 70� 1
16�� 116� � 0.2734
109.
(a)
(b)
0−5 18
g
f
600
� 0.064t2 � 6.74t � 256.1, �15 ≤ t ≤ 3
� 0.064�t � 20�2 � 9.30�t � 20� � 416.5
g�t� � f�t � 20�
f�t� � 0.064t2 � 9.30t � 416.5, 5 ≤ t ≤ 23 110.
(a)
(b)
0−5 19
15,000
g
f
� 6.22t2 � 364t � 7522, �15 ≤ t ≤ 5
� 6.22�t � 20�2 � 115.2�t � 20� � 2730
g�t� � f�t � 20�
f �t� � 6.22t2 � 115.2 t � 2730, 5 ≤ t ≤ 25
111. False. The term is
�Note: 7920 is the coefficient of x8y4.�12C4x
4��2y�8 � 495x4��2�8y8 � 126,720x4y8.
x4y8 112. False. The coefficient of is and thecoefficient of is 192,456.x14
1,732,104x10
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Section 8.5 The Binomial Theorem 747
113. Answers will vary. See page 557.
114. Rows 8–10 of Pascal’s Triangle are:
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
115. The expansions of and arealmost the same except that the signs of the terms in the expansion of alternate from positive to negative.
�x � y�n
�x � y�n�x � y�n
117.
�n!
r!�n � r�!�
n!
�n � r�!r!� nCr
nCn�r �n!
�n � �n � r��!�n � r�!
116. (a) Second term of is
(b) Fourth term of is
6C3�12 x�3�7y�3 � 857.5x3y3.
�12 x � 7y�6
5�2x�4��3y�1 � �240x4y.
�2x � 3y�5
118.
� 0
� nC0 � nC1 � nC2 � nC3 � . . . �±nCn �
0 � �1 � 1�n
119.
��n � 1�!
�n � 1 � r�!r!� n�1Cr
�n!�n � 1�
�n � r � 1�!r!
�n!�n � r � 1 � r�
�n � r � 1�!r!
�n!�n � r � 1��n � r � 1�!r!
�n!r
�n � r � 1�!r!
�n!�n � r � 1�
�n � r�!r!�n � r � 1��
� � r � 1�n!�r � 1�!r�n � r � 1�!�r � 1�!r
nCr � nCr�1 �n!
�n � r�!r!�
n!
�n � r � 1�!�r � 1�!
120. nC0 � nC1 � nC2 � nC3 � . . . � nCn � �1 � 1�n � 2n
122.
is shifted three units to the right of f�x�.g�x�
g�x� � f �x � 3�
124.
is the reflection of in the -axis.xf�x�g�x�
g�x� � �f�x�
121.
is shifted eight units up from .f�x�g�x�
g�x� � f�x� � 8
123.
is the reflection of in the y-axis.f�x�g�x�
g�x� � f��x�
126. 1.2�2
�2.34�
�1
�1
4.8 � 4.6 4
22.31.2� � 20
1011.5
6�
125. �6�5
54�
�1
�1
�24 � 2545
�5�6� � 4
5�5�6�©
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748 Chapter 8 Sequences, Series, and Probability
Section 8.6 Counting Principles
■ You should know The Fundamental Counting Principle.
■ is the number of permutations of n elements taken r at a time.
■ Given a set of n objects that has of one kind, of a second kind, and so on, the number of distinguishablepermutations is
■ is the number of combinations of n elements taken r at a time.nCr �n!
�n � r�!r!
n!
n1!n2! . . . nk!.
n2n1
nPr �n!
�n � r�!
1. Odd integers: 1, 3, 5, 7, 9, 11
6 ways
3. Prime integers: 2, 3, 5, 7, 11
5 ways
5. Divisible by 4: 4, 8, 12
3 ways
7. Sum is 8:
7 ways
1 � 7, 2 � 6, 3 � 5, 4 � 4, 5 � 3, 6 � 2, 7 � 1
9. Amplifiers: 4 choices
Compact disc players: 6 choices
Speakers: 5 choices
Total: ways4 � 6 � 5 � 120
2. Even integers: 2, 4, 6, 8, 10, 12
6 ways
4. Greater than 6: 7, 8, 9, 10, 11, 12
6 ways
6. Divisible by 3: 3, 6, 9, 12
4 ways
8. Distinct integers whose sum is 8:
6 ways
3 � 5, 5 � 3, 6 � 2, 7 � 12 � 6,1 � 7,
Vocabulary Check
1. Fundamental Counting Principle 2. permutation
3. 4. distinguishable permutations
5. combinations
nPr �n!
�n � r�!
10. Math courses: 2
Science courses: 3
Social sciences and humanities courses: 5
Total: 2 � 3 � 5 � 30 ways
11. ways210 � 1024
12. First lock:
Second lock:
Hence,combinations.106 � 1,000,000
10 � 10 � 10
10 � 10 � 10 13. (a)
(b) 9 � 9 � 8 � 648
9 � 10 � 10 � 900 14. (a)
(b) 9 � 10 � 10 � 5 � 4500
4 � 10 � 10 � 10 � 4000
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Section 8.6 Counting Principles 749
37.
� 11,880 ways
� 12 � 11 � 10 � 9
12P4 �12!8!
38.
� 1,816,214,400 ways
� 15 � 14 � 13 � 12 � 11 � 10 � 9 � 8 � 7
15P9 �15!6!
39. 37 � 37 � 37 � 50,653 40. 8P3 �8!5!
� 336 orders
41. ABCD BACD CABD DABC
ABDC BADC CADB DACB
ACBD BCAD CBAD DBAC
ACDB BCDA CBDA DBCA
ADBC BDAC CDAB DCAB
ADCB BDCA CDBA DCBA
42. ABCD
ACBD
DBCA
DCBA
15. numbers2�8 � 10 � 10��10 � 10 � 10 � 10� � 16,000,000 16. telephone numbers4�8,000,000� � 32,000,000
22. (a) ways
(b) ways�5!��3!� � 120�6� � 720
8! � 40,320
17. (a)
(b) There are possibilities that don’t haveQ. Hence, have atleast one Q.
2 � 263 � 2 � 253 � 39022 � 253
263 � 263 � 35,152 18. (a) ATM codes
(b) ATM codes that don’t beginwith zero9 � 103 � 9000
104 � 10,000
19. (a) zip codes
(b) zip codes beginning with aone or a two2 � 104 � 20,000
105 � 100,000 20. (a) nine-digit zip codes
(b) nine-digit zip codes beginning with aone or a two2 � 108
109
21. (a)
(b) 6 � 1 � 4 � 1 � 2 � 1 � 48
6 � 5 � 4 � 3 � 2 � 1 � 720
23.
So, 4P4 �4!
0!� 4! � 24.
nPr �n!
�n � r�!24.
5P5 �5!
�5 � 5�!�
5!
0!� 120
nPr �n!
�n � r�!
26. 20P2 �20!
18!� 20�19� � 380
25. 8P3 �8!
5!� 8 � 7 � 6 � 336
27. 5P4 �5!
1!� 120
29. 20P6 � 27,907,200 31. 120P4 � 197,149,680
28.
� 7 � 6 � 5 � 4 � 840
7P4 �7!
3!
32. 100P5 � 9,034,502,400
30. 10P8 � 1,814,400
33. 5! � 120 ways 34. 4! � 24
35. ways9! � 362,880 36. 4! � 24 ways
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750 Chapter 8 Sequences, Series, and Probability
43.7!
2!1!3!1!�
7!
2!3!� 420
45.
� 2520
� 7 � 6 � 5 � 4 � 3
7!
2!1!1!1!1!1!�
7!
2!
44.8!
3!5!� 56
46.11!
1!4!4!2!�
11!
4!4!2!� 34,650 47. 5C2 �
5!2!3!
�5 � 4
2� 10
48. 6C3 �6!
3!3!�
6 � 5 � 46
� 20 49. 4C1 �4!
1!3!� 4 50. 5C1 �
5!1!4!
� 5
51. 25C0 �25!
0!25!� 1 52. 20C0 �
20!0!20!
� 1
54. 10C7 � 120
53. 20C4 � 4845
55. 42C5 � 850,668 56. 50C6 � 15,890,700
57. AB, AC, AD, AE, AF,BC, BD, BE, BF, CDCE, CF, DE, DF, EF
ways6C2 � 15
58. ABC, ABD, ABE, ABF, ACD, ACE, ACF ADE, ADF, AEF, BCD, BCE, BCF, BDEBDF, BEF, CDE, CDF, CEF, DEF
ways6C3 � 20
59. 100C14 �100!
14!86!� 4.42 � 1016 ways 60. ways14C12 � 91
61. 49C6 � 13,983,816 ways 62.
� 146,107,962 combinations
�55C5��42C1� � �3,478,761��42�
63. lines9C2 � 36
64. There are 22 good sets and 3 defective sets.
(a) ways
(b) ways
(c) ways22C4 � �22C3��3C1� � �22C2 ��3C2 � � 7315 � �1540��3� � 693 � 12,628
�22C2 ��3C2 � � �231��3� � 693
22C4 � 7315
65. Select type of card for three of a kind:
Select three of four cards for three of a kind:
Select type of card for pair:
Select two of four cards for pair:
ways to get a full house13C1 � 4C3 � 12C1 � 4C2 � 13 � 4 � 12 � 6 � 3744
4C2
12C1
4C3
13C1
67. (a)
(b) �5C2 ��7C2 � � �10��21� � 210 ways
12C4 � 495 ways66. Select 2 jacks:
Select 3 aces:
Total: ways6 � 4 � 24
4C3 � 4
4C2 � 6
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Section 8.6 Counting Principles 751
68. ways�13C7��20C3� � 1716 � 1140 � 1,956,240 69.
� 292,600 ways
�7C1��12C3��20C2� � 7 � 220 � 190
71. 5C2 � 5 � 10 � 5 � 5 diagonals
70. (a) relationships
(c) relationships12C2 �12!
2!10!�
12 � 11
2� 66
3C2 �3!
2!1!� 3 (b) relationships
(d) relationships20C2 �20!
2!18!�
20 � 19
2� 190
8C2 �8!
2!6!�
8 � 7
2� 28
72. diagonals6C2 � 6 � 15 � 6 � 9
74. diagonals10C2 � 10 � 45 � 10 � 3573. 8C2 � 8 � 28 � 8 � 20 diagonals
75.
Note: for this to be defined.
n � 5 or n � 6
0 � �n � 5��n � 6�
0 � n2 � 11n � 30
14n � 28 � n2 � 3n � 2
�We can divide here by n�n � 1� since n � 0, n � 1.� 14n�n � 1��n � 2� � �n � 2��n � 1�n�n � 1�
14� n!
�n � 3�!� ��n � 2�!�n � 2�!
n ≥ 3
14 � nP3 � n�2P4
76.
Note: for this to be defined.
n � 9 or n � 10
�n � 9��n � 10� � 0
n2 � 19n � 90 � 0
n2 � n � 18n � 90
n�n � 1��n � 2��n � 3��n � 4� � 18�n � 2��n � 3��n � 4��n � 5�
n!
�n � 5�!� 18��n � 2�!
�n � 6�!�n ≥ 6
nP5 � 18 � n�2P4
� We can divide by �n � 2�, �n � 3�, �n � 4� since n � 2, n � 3, and n � 4.�
77.
n � 10
n! � 10�n � 1�!
n!
�n � 4�! � 10�n � 1�!�n � 4�!
nP4 � 10 � n�1P3 78.
n � 12
n! � 12�n � 1�!
n!
�n � 6�! � 12�n � 1�!�n � 6�!
nP6 � 12 � n�1P5 79.
n � 3
�n � 1�! � 4n!
�n � 1�!�n � 2�! � 4
n!�n � 2�!
n�1P3 � 4 � nP2
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752 Chapter 8 Sequences, Series, and Probability
83. False
85.
This number is too large for some calculators to evaluate.
100P80 � 3.836 � 10139.
87. nCr � nCn�r �n!
r!�n � r�!
84. True
86. The symbol means the number of ways to choose and order elements out of a set of
elements.nr
nPr
88. (b) is larger than because thepermutations count different orderingsas distinct.
10C610P6
89. nPn�1�n!
�n � �n � 1��!�
n!
1!�
n!
0!� nPn
90.
�n!
n!0!�
n!
�n � 0�!0!� nC0
�n!
0!n!
nCn �n!
�n � n�!n!91.
�n!
�n � 1�!1!� nC1
�n!
�1�!�n � 1�!
nCn�1 �n!
�n � �n � 1�!�n � 1�!92.
� nPr
r!
�1r!�
n!�n � r�!�
nCr �n!
�n � r�!r!
80.
n � 2
�n � 1��n� � 6
�n � 1�! � 6�n � 1�!
�n � 2�!�n � 1�! � 6
�n � 2�!�n � 1�!
n�2P3 � 6 � n�2P1 81.
n � 2
4�n � 1�! � �n � 2�!
4�n � 1�!�n � 1�! �
�n � 2�!�n � 1�!
4 � n�1P2 � n�2P3 82.
n � 5
5�n � 1�! � n!
5�n � 1�!�n � 2�! �
n!�n � 2�!
5 � n�1P1 � nP2
94.
t �11
2� 5.5
11 � 2t
8 � 3
2t� 1
4
t�
3
2t� 1
93. From the graph of you see that there is one zero, Analytically,
By the Quadratic Formula,
Selecting the larger solution, (The other solution is extraneous.)x �13 � 13
2� 8.303.
x �13 ± ��13�2 � 4�39�
2�
13 ± 13
2.
0 � x2 � 13x � 39.
x � 3 � x2 � 12x � 36
x � 3 � x � 6
x � 8.303.y � x � 3 � x � 6,
95.
x � 35
25 � 3 � x
25 � x � 3
log2�x � 3� � 5 96.
x � 3 ln 16 � 8.318
x
3� ln 16
ex�3 � 16
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Section 8.7 Probability 753
You should know the following basic principles of probability.
■ If an event E has equally likely outcomes and its sample space has equally likely outcomes, then theprobability of event E is
where
■ If A and B are mutually exclusive events, then
If A and B are not mutually exclusive events, then
■ If A and B are independent events, then the probability that both A and B will occur is
■ The probability of the complement of an event A is P�A�� � 1 � P�A�.P�A�P�B�.
P�A � B� � P�A� � P�B� � P�A � B�.P�A � B� � P�A� � P�B�.
0 ≤ P�E� ≤ 1.P�E� �n�E�n�S�
,
n�S�n�E�
Section 8.7 Probability
98.
Answer: �6, �13�
y ��86 35
10��86 1
2� ��130
10� �13
x ��3510
12�
�86 12� �
6010
� 697.
Answer: ��2, �8�
y ���5
7�14
2���5
73
�2� �88
�11� �8
x ���14
23
�2���5
73
�2� �22
�11� �2
99.
Answer: ��1, 1�
y ���3
9�1�4�
��39
�45� �
2121
� 1
x ���1�4
�45�
��39
�45� �
�2121
� �1 100.
Answer: ��3, 4�
y �� 10�8
�748�
� 10�8
�11�4� �
�512�128
� 4
x ���74
8�11�4�
� 10�8
�11�4� �
384�128
� �3
Vocabulary Check
1. experiment, outcomes 2. sample space 3. probability
4. impossible, certain 5. mutually exclusive 6. independent
7. complement 8. (a) iii (b) i (c) iv (d) ii
1. �T, 1�, �T, 2�, �T, 3�, �T, 4�, �T, 5�, �T, 6����H, 1�, �H, 2�, �H, 3�, �H, 4�, �H, 5�, �H, 6�,
3. �ABC, ACB, BAC, BCA, CAB, CBA�
2. �2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12�
4.�Y, B�, �Y, R����R, R�, �R, B�, �R, Y�, �B, B�, �B, Y�, �B, R�,
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754 Chapter 8 Sequences, Series, and Probability
5.�B, D�, �B, E�, �C, D�, �C, E�, �D, E����A, B�, �A, C�, �A, D�, �A, E�, �B, C�,
7.
P�E� �n�E�n�S�
�3
8
E � �HTT, THT, TTH�
9.
P�E� �n�E�n�S�
�7
8
E � �HHH, HHT, HTH, HTT, THH, THT, TTH�
6. �SSS, SSF, SFS, FSS, SFF, FFS, FSF, FFF�
8.
P�E� �n�E�n�S�
�4
8�
1
2
E � �HHH, HHT, HTH, HTT�
10.
P�E� �n�E�n�S�
�4
8�
1
2
E � �HHH, HHT, HTH, THH�
11.
P�E� �n�E�n�S�
�12
52�
3
13
E � �K, K, K, K, Q, Q, Q, Q, J, J, J, J� 12. The probability that the card is not a black facecard is the complement of getting a black face card.
Hence, and
P�E� � � 1 � P�E� � 1 �652 �
2326.
P�E� �652
E � �K, K, Q, Q, J, J�
13.
P�E� �n�E�n�S� �
1652
�413
E � �A, A, A, A, K, K, K, K, Q, Q, Q, Q, J, J, J, J� 14. There are 9 possible cards in each of 4 suits.
P�E� �n�E�n�S � �
3652
�913
9 � 4 � 36
15.
P�E� �n�E�n�S� �
536
E � ��1, 5�, �2, 4�, �3, 3�, �4, 2�, �5, 1�� 16.
P�E� �n�E�n�S�
�15
36�
5
12
�6, 5�, �6, 6���5, 3�, �5, 4�, �5, 5�, �5, 6�, �6, 2�, �6, 3�, �6, 4�,E � ��2, 6�, �3, 5�, �3, 6�, �4, 4�, �4, 5�, �4, 6�,
18.
P�E� �n�E�n�S�
�19
36
�5, 2�, �5, 4�, �5, 6�, �6, 1�, �6, 3�, �6, 5���2, 5�, �3, 2�, �3, 4�, �3, 6�, �4, 1�, �4, 3�, �4, 5�,E � ��1, 1�, �1, 2�, �1, 4�, �1, 6�, �2, 1�, �2, 3�,17. not
P�E� �n�E�n�S� �
3336
�1112
n�E� � n�S� � n�not E� � 36 � 3 � 33
E � ��5, 6�, �6, 5�, �6, 6��
19. P�E� � 3C2
6C2
�3
15�
1
5
21. P�E� � 4C2
6C2
�6
15�
2
5
20. P�E� � 2C2
6C2
�1
15
22.
�2 � 3 � 6
15�
11
15
P�E� � 1C1 � 2C1 � 1C1 � 3C1 � 2C1 � 3C1
6C2
23. P�E�� � 1 � P��� � 1 � 0.75 � 0.25 24. P�E�� � 1 � P�E� � 1 � 0.2 � 1 �29 �
79 � 0.7
25. P�E�� � 1 � P�E� � 1 �23 �
13 26. P�E�� � 1 � P�E� � 1 �
78 �
18
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Section 8.7 Probability 755
35. (a)
(b)
(c)23100
� 0.23
45100
� 0.45
34100
� 0.34 37. (a)
(b)
(c)672 � 124
1254�
548
1254
582
1254
672
1254
39.
Taylor: Moore:
Perez: 0.25 �1
4
0.25 �1
40.50 �
1
2,
p � 0.25
p � p � 2p � 1
36. (a)
(b)
(c)2
500� 0.004
478
500� 0.956
290
500� 0.58
38. (a)
(b)
(c)4
128�
132
4 � 20128
�24128
�316
�Note: 1 �1316
�316�
48 � 56128
�104128
�1316
40.54
31 � 54 � 42 � 20 � 47 � 58�
54252
�314 41. (a)
(b)
(c)
�28,028
184,756�
49
323� 0.152
15C9 � 5C1
20C10
� 15C10
20C10
�25,025 � 3003
184,756
15C8 � 5C2
20C10
�64,350
184,756�
225
646� 0.348
15C10
20C10
�3003
184,756�
21
1292� 0.016
27. P�E� � 1 � P�E� � � 1 � p � 1 � 0.12 � 0.88
29. P�E� � 1 � P�E� � � 1 �1320 �
720
28. 1 � p � 1 � 0.84 � 0.16P�E� � 1 � P�E�� �
30. P�E� � 1 � P�E�� � 1 �61100 �
39100
31. (a)
(b)
(c)
(d)0.24 � 0.02
1.0� 0.26
0.241.0
� 0.24
0.411.0
� 0.41
0.15�8.15� � 1.22 million 32. (a) presidents had no children.
(b) presidents had four children.
(c)
(d)
Answers will vary.
0.14
�0.07 � 0.21� � 0.28
0.19�42� � 8
0.10�42� � 4
33. (a)
(b)
(c)0.01 � 0.002
1.0� 0.012
0.011.0
� 0.01
�0.128��293.66� � 37.6 million 34. (a) probability of having a highschool diploma. Hence,
(b)
(c)
(d)
(e) 0.084 � 0.181 � 0.097 � 0.362
1 � 0.148 � 0.852
0.181 � 0.097 � 0.278
0.097�186.88� � 18.1 million
0.852�186.88� � 159.2 million.
1 � 0.148 � 0.852
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756 Chapter 8 Sequences, Series, and Probability
42. Total ways to insert paychecks: ways
5 correct: 1 way
4 correct: not possible
3 correct: 10 ways
2 correct: 20 ways
1 correct: 45 ways
0 correct: 44 ways
(a) (b)45 � 20 � 10 � 1
120�
19
30
45
120�
3
8
5! � 120
43. (a)
(b)1
4P4
�1
24
1
5P5
�1
12044. (a)
(b)�8C2 ��25C2 ��25C3�
108C7� 6.929 � 10�4
�8C2 ��100C5�108C7
� 0.0756
45. (a) There are three letters to be selected, and twomust be Q and Y.
QY__, YQ__, Q__Y, Y__Q, __YQ, __QY
Thus, the probability is
6�26�263 �
6262 � 0.008876.
(b) The three letters must be Q, Y, and X.
QYX, QXY, YQX, YXQ, XQY, XYQ
Thus, the probability is 6
263 �3
8788.
49. (a)
(b)
(c)4 � 12
52�
413
13 � 1352
�12
2052
�513 50.
�6
4165
�3744
2,598,960
13C1 � 4C3 � 12C1 � 4C2
52C5
�13 � 4 � 12 � 6
2,598,960
51. (a) (4 good units)
(b) (2 good units)
(c) (3 good units)
At least 2 good units:12
55�
28
55�
14
55�
54
55
�9C3��3C1�12C4
�252495
�2855
�9C2��3C2�12C4
�108495
�1255
9C4
12C4
�126
495�
14
5552. (a)
(b)
(c)
(d) P�N1N1� �40
40�
1
40�
1
40
P�N1 < 30, N2 < 30� �29
40�
29
40�
841
1600
P�EO or OE� � 220
4020
40 �1
2
P�EE� �20
40�
20
40�
1
4
46. (a)
(b)1
102 � 0.01
1104 � 0.0001 47. (a)
(b)1000
�55C5��42C1� �
1000�3,478,761��42�
100�55C5��42C1�
�100
�3,478,761��42� 48. (a)
(b)
(c)1
102
1104
1109
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Section 8.7 Probability 757
63. (a) As you consider successive people with distinct birthdays, the probabilities mustdecrease to take into account the birth dates already used. Since the birth dates of peopleare independent events, multiply the respective probabilities of distinct birthdays.
(b)
(c)
(d) is the probability that the birthdays are not distinct which is equivalent to at least 2 people having the same birthday.
—CONTINUED—
Qn
Pn �365
365�
364
365�
363
365� . . . �
365 � �n � 1�365
�365 � �n � 1�
365Pn�1
P3 �365
365�
364
365�
363
365�
363
365 P2 �
365 � �3 � 1�365
P2
P2 �365
365�
364
365�
364
365 P1 �
365 � �2 � 1�365
P1
P1 �365
365� 1
365
365�
364
365�
363
365�
362
365
53. �0.32�2 � 0.1024 54. �0.78�3 � 0.474552
55. (a)
(b)
(c) P�FF� � �0.015�2 � 0.0002
P�S� � 1 � P�FF� � 1 � �0.015�2 � 0.9998
P�SS� � �0.985�2 � 0.9702
57. (a)
(b)
(c) 1 � 0.262144 � 0.737856 �11,52915,625
45
6
�4096
15,625� 0.262144
15
6
�1
15,625
56. (a)
(b)
(c) P�A� � 1 � P�NN� � 1 � 0.01 � 0.99
P�NN� � �0.10�2 � 0.01
P�AA� � �0.90�2 � 0.81
58. (a)
(b)
(c)
� 1 �1
16�
1516
� 1 � P�GGGG�P�at least one boy� � 1 � P�no boys�
P�BBBB� � P�GGGG� � 1
24
� 1
24
�1
8
P�BBBB� � 1
24
�1
16
59. (a) If the center of the coin falls within the circle of radius around a vertex,the coin will cover the vertex.
(b) Experimental results will vary.
�
n��d
22
�nd2
�1
4�P�coin covers a vertex� �
Area in which coin may fallso that it covers a vertex
Total area
d�2
60. � 1 � 3
42
� 1 �9
16�
7
161 �
�45�2
�60�2� 1 � 45
602
61. True
P�E� � P�E� � � 1
62. False. The first sentence is true, but the second isfalse. The complement is to roll a number greaterthan 2, and its probability is 2
3.
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758 Chapter 8 Sequences, Series, and Probability
64. If a weather forecast indicates that the probability of rain is 40%, this means themeteorological records indicate that over an extended period of time with similarweather conditions it will rain 40% of the time.
65.
x �112
4x � 22
2 � 4�x � 5� � 4x � 20
2
x � 5� 4
67.
x � �10
3x � 6 � x2 � 2x � x2 � 4
3�x � 2� � x�x � 2� � �x � 2��x � 2�
3
x � 2�
xx � 2
� 1
66.
x � �1
2x � �2
1 � 2x � 3
4
2x � 3� 4
3
2x � 3� 4 �
�12x � 3
68.
x � 3
�3x � �9
2�x � 2� � 5�x� � �13
2x
�5
x � 2�
�13x2 � 2x
��13
x�x � 2�
69.
x � ln�28� � 3.332
ex � 28
ex � 7 � 35 70.
x � �ln�38� � ln�8
3� � 0.981
�x � ln�38�
e�x �75200 �
38
200e�x � 75
63. —CONTINUED—
(e)
(f) 23, See the chart above.
n 10 15 20 23 30 40 50
0.88 0.75 0.59 0.49 0.29 0.11 0.03
0.12 0.25 0.41 0.51 0.71 0.89 0.97Qn
Pn
71.
x �16e4 � 9.10
e4 � 6x
ln 6x � 4
4 ln 6x � 16 72.
x �12 e3 � 10.043
2x � e3
ln 2x � 3
5 ln 2x � 4 � 11
73. 5P3 �5!
�5 � 3�! �1202
� 60
75. 11P8 �11!
�11 � 8�! �11!3!
� 6,652,800
77. 6C2 �6!
4!2!�
6 � 5 � 4!4!2
� 15
74. 10P4 �10!
�10 � 4�! �10!6!
� 10 � 9 � 8 � 7 � 5040
76. 9P2 �9!
�9 � 2�! �9!7!
� 9 � 8 � 72
78. 9C5 � 126
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Review Exercises for Chapter 8 759
7. Denominators are successiveodd numbers.
an �2
2n � 1, n � 1, 2, 3, . . .
8. an �n � 2n � 1
, n � 1, 2, 3, . . . 9.
a5 � �3 � 4 � �7
a4 � 1 � 4 � �3
a3 � 5 � 4 � 1
a2 � a1 � 4 � 9 � 4 � 5
a1 � 9, ak�1 � ak � 4
10.
a5 � 73
a4 � 67
a3 � 55 � 6 � 61
a2 � a1 � 6 � 49 � 6 � 55
a1 � 49, ak�1 � ak � 6 11.
�1
20 � 19�
1
380
18!
20!�
18!
20 � 19 � 18!12.
10!8!
�10 � 9 � 8!
8!� 90
Review Exercises for Chapter 8
1.
a5 �25
25 � 1�
3233
a4 �24
24 � 1�
1617
a3 �23
23 � 1�
89
a2 �22
22 � 1�
45
a1 �21
21 � 1�
23
an �2n
2n � 12.
a5 �15
�16
�130
a4 �14
�15
�120
a3 �13
�14
�112
a2 �12
�13
�16
a1 �11
�12
�12
an �1n
�1
n � 13.
a5 ���1�5
5!� �
1120
a4 ���1�4
4!�
124
a3 ���1�3
3!� �
16
a2 ���1�2
2!�
12
a1 ���1�1
1!� �1
an ���1�n
n!
4.
a5 ���1�5
11!�
�139,916,800
a4 ���1�4
9!�
1362,880
a3 ���1�3
7!�
�15040
a2 ���1�2
5!�
1120
a1 ���1�1
3!� �
16
an ���1�n
�2n � 1�! 5. Common difference is 5.
an � 5n, n � 1, 2, . . .
6. Common difference is
an � 52 � 2n, n � 1, 2, 3, . . .
�2.
79. 11C8 �11!8!3!
�11 � 10 � 9 � 8!
8!6� 165 80. 16C13 � 560
©H
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760 Chapter 8 Sequences, Series, and Probability
15. �6
i�1
5 � 6�5� � 30
17.
� 6 �3
2�
2
3�
3
8�
205
24
�4
j�1
6
j2�
6
12�
6
22�
6
32�
6
42
16. �5
k�2
4k � 8 � 12 � 16 � 20 � 56
18.
� 6.17
� 8
i�1
i
i � 1�
1
2�
2
3�
3
4�
4
5�
5
6�
6
7�
7
8�
8
9
19. �100
k�12k3 � 2 �
1002�101�2
4� 51,005,000 20. �
40
j�0� j2 � 1� �
40�41��81�6
� 41 � 22,181
21. �50
n�0�n2 � 3� �
50�51��101�6
� 3�51� � 43,078
22.
�1
1�
1
101�
100
101
�100
n�1�1
n�
1
n � 1� � �1
1�
1
2� � �1
2�
1
3� � �1
3�
1
4� � . . . � � 1
99�
1
100� � � 1
100�
1
101�
23.
� 1.799
1
2�1��
1
2�2��
1
2�3�� . . . �
1
2�20�� �
20
k�1
1
2k24.
� 570
2�12� � 2�22� � 2�32� � . . . � 2�92� � �9
k�1
2k2
25.1
2�
2
3�
3
4� . . . �
9
10� �
9
k�1
k
k � 1� 7.071 26. 1 �
1
3�
1
9�
1
27� . . . � �
�
k�0��
1
3�k
�3
4
13.�n � 1�!�n � 1�! �
�n � 1�n�n � 1�!�n � 1�! � n�n � 1� 14.
2n!�n � 1�! �
2n!�n � 1�n!
�2
n � 1
27. (a)
(b) ��
k�1
510k �
510
��
k�0
110k �
510
�1
1 � 1�10�
510
�109
�59
�4
k�1
510k �
510
�5
100�
51000
�5
10,000� 0.5 � 0.05 � 0.005 � 0.0005 � 0.5555 �
11112000
28.
(a)
(b) ��
k�1
32k � �
�
k�0�3
2��12k� �
3�21 � 1�2
� 3
�4
k�1
32k �
32
�34
�38
�316
�4516
� 2.8125
��
k�1
32k 29.
(a)
(b) ��
k�12�0.5�k � 2�0.5� 1
1 � 0.5� 2
� 1.875 �158
�4
k�12�0.5�k � 2�0.5� � 2�0.5�2 � 2�0.5�3 � 2�0.5�4
��
k�12�0.5�k
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Miff
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Review Exercises for Chapter 8 761
37.
a5 � 15 � 4 � 19
a4 � 11 � 4 � 15
a3 � 7 � 4 � 11
a2 � 3 � 4 � 7
a1 � 3
a1 � 3, d � 4 39.
a5 � 10 � 3 � 13
a4 � 7 � 3 � 10
a3 � 4 � 3 � 7
a2 � 1 � 3 � 4
a1 � 1
a1 � 10 � 3�3�
a1 � a4 � 3d
13 � d
18 � 6d
28 � 10 � 6d
a10 � a4 � 6d
a4 � 10, a10 � 2838.
a5 � 2 � 2 � 0
a4 � 4 � 2 � 2
a3 � 6 � 2 � 4
a2 � 8 � 2 � 6
a1 � 8
a1 � 8, d � �2 40.
a5 � 18 � 2 � 20
a4 � 16 � 2 � 18
a3 � 14 � 2 � 16
a2 � 12 � 2 � 14
a1 � 14 � 2 � 12
a1 � a2 � d
2 � d
8 � 4d
22 � 14 � 4d
a6 � a2 � 4d
a2 � 14, a6 � 22
30.
(a)
(b) ��
k�14�0.25�k � �
�
k�04�0.25��0.25�k �
11 � 0.25
�43
��
k�14�0.25�k � 4�0.25� � 4�0.25�2 � 4�0.25�3 � 4�0.25�4 � 1.328125 �
8564
��
k�14�0.25�k
31.
(a) (b)
a8 � 2601.77
a7 � 2588.82a6 � 2575.94
a5 � 2563.13a4 � 2550.38
a3 � 2537.69a2 � 2525.06
a40 � 2500�1 �0.02
4 �40
� $3051.99a1 � 2500�1 �0.02
4 �1
� 2512.5
an � 2500�1 �0.02
4 �n
, n � 1, 2, 3
32. (a) (b)
(c) (d) For 2004, and thousand.
For 2010, and thousand.
The results seem reasonable.
a20 � 750n � 20
a14 � 650n � 14
0 14500
700
0 14500
7001 2 3 4 5 6
535 539 544 549 556 563an
n
7 8 9 10 11 12 13
571 580 590 600 611 623 636an
n
33. Yes
d � 3 � 5 � �2
34. Not arithmetic 35. Yes
d � 1 �12
�12
36. Arithmetic
d �89
�99
��19
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762 Chapter 8 Sequences, Series, and Probability
42.
d �52an � 15 �
52�n � 1� �
252 �
52n,
a5 �452 �
52 �
502 � 25
a4 � 20 �52 �
452
a3 �352 �
52 �
402 � 20
a2 � 15 �52 �
352
a1 � 15
a1 � 15, ak�1 � ak �5241.
an � 35 � �n � 1���3� � 38 � 3n, d � �3
a5 � a4 � 3 � 26 � 3 � 23
a4 � a3 � 3 � 29 � 3 � 26
a3 � a2 � 3 � 32 � 3 � 29
a2 � a1 � 3 � 35 � 3 � 32
a1 � 35
a1 � 35, ak�1 � ak � 3
43.
an � 9 � �n � 1��7� � 2 � 7n, d � 7
a5 � a4 � 7 � 30 � 7 � 37
a4 � a3 � 7 � 23 � 7 � 30
a3 � a2 � 7 � 16 � 7 � 23
a2 � a1 � 7 � 9 � 7 � 16
a1 � 9
a1 � 9, ak�1 � ak � 7 44.
d � �5 an � 100 � 5�n � 1� � 105 � 5n,
a5 � 85 � 5 � 80
a4 � 90 � 5 � 85
a3 � 95 � 5 � 90
a2 � 100 � 5 � 95
a1 � 100
a1 � 100, ak�1 � ak � 5
45.
�20
n�1
�103 � 3n� � �20
n�1
103 � 3 �20
n�1
n � 20�103� � 3��20��21�2 � 1430
an � 100 � �n � 1���3� � 103 � 3n
47.
� 2�10�11�2 � 10�3� � 80
�10
j�1
�2j � 3� � 2�10
j�1 j � �
10
j�1
3
49.
�2
3�
�11��12�2
� 11�4� � 88
�11
k�1�2
3k � 4� �
2
3 �11
k�1
k � �11
k�1
4
48.
� 8�20� � 3��8��9�2 � 52
�8
j�1
�20 � 3j� � �8
j�1
20 � 3� 8
j�1
j
50.
�3
4��25��26�
2 � 25�1
4� � 250
�25
k�1�3k � 1
4 � �3
4 �25
k�1 k � �
25
k�1
1
4
46.
� 20�1� � 9��20��21�2 � 1910 �
20
n�1
�1 � 9n� � �20
n�1
1 � 9 �20
n�1
n
an � 10 � �n � 1�9 � 1 � 9n
9 � d
18 � 2d
28 � 10 � 2d
a3 � a1 � 2d
51. �100
k�1
5k � 5��100��101�2 � 25,250 52.
� 3050
�80
n�20
n � �80
n�1
n � �19
n�1
n ��80��81�
2�
�19��20�2
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.
Review Exercises for Chapter 8 763
62.
±6 � r
6 � r2
12 � 2r2
a3 � a1r2
a1 � 2, a3 � 12
a5 � �126��6� � 72
a4 � 12��6� � �126
a3 � �26��6� � 12
a2 � 2��6� � �26
a1 � 2
or
a5 � 126 6 � 72
a4 � 12�6� � 126
a3 � 26�6� � 12
a2 � 2�6� � 26
a1 � 2
63.
an � 120�13�n�1
, r �13
a5 �13�40
9 � �4027
a4 �13�40
3 � �409
a3 �13�40� �
403
a2 �13�120� � 40
a1 � 120
a1 � 120, ak�1 �13ak 65.
an � 25��35�n�1
, r � �35
a5 � �35��27
5 � �8125
a4 � �35�9� � �
273
a3 � �35��15� � 9
a2 � �35�25� � �15
a1 � 25
a1 � 25, ak�1 � �35ak64.
an � 200�0.1�n�1
a5 � 0.1�0.2� � 0.02
a4 � 0.1�2� � 0.2
a3 � 0.1�20� � 2
a2 � 0.1�200� � 20
a1 � 200
a1 � 200, ak�1 � 0.1ak
61.
49 � r2 ⇒ r � ±2
3
4 � 9r2
a3 � a1r2
a1 � 9, a3 � 4
or
a5 � �83��2
3� �169a5 �
83�2
3� �169
a4 � 4��23� � �
83a4 � 4�2
3� �83
a3 � �6��23� � 4a3 � 6�2
3� � 4
a2 � 9��23� � �6a2 � 9�2
3� � 6
a1 � 9a1 � 9
53. (a)
(b)
� $192,500
� �5
k�1
�31,750 � 2250k�
�5
k�1
�34,000 � �k � 1��2250��
34,000 � 4�2250� � $43,000 54.
bales S8 �82�123 � 46� � 676
a8 � ��11�8 � 134 � 46
n � 8
a1 � 123, d � 112 � 123 � �11
55. 5, 10, 20, 40
Geometric: r � 2
56.
Not geometric:
23r �
34 ⇒ r �
98
12r �
23 ⇒ r �
43
12, 23, 34, 45 57. Geometric:
r � �13
58. Geometric:
r � �2
59.
a5 � �116��1
4� �164
a4 �14��1
4� � �116
a3 � �1��14� �
14
a2 � 4��14� � �1
a1 � 4
a1 � 4, r � �14 60.
a5 �274 �3
2� �818
a4 �92�3
2� �274
a3 � 3�32� �
92
a2 � 2�32� � 3
a1 � 2
a1 � 2, r �32
©H
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.
764 Chapter 8 Sequences, Series, and Probability
66.
r �53
an � 18�5
3�n�1
a5 �5
3�250
3 � �1250
9
a4 �5
3�50� �
250
3
a3 �5
3�30� � 50
a2 �5
3�18� � 30
a1 � 18
a1 � 18, ak�1 �5
3ak 67.
�20
n�1
16��1
2�n�1
� 16�1 � ��1�2�20
1 � ��1�2� � 10.67
an � 16��1
2�n�1
�1
2� r
�8 � 16r
a2 � a1r
69.
�20
n�1
100�1.05�n�1 � 100�1 � �1.05�20
1 � 1.05 � 3306.60
an � 100�1.05�n�1
a1 � 100, r � 1.0568.
�20
n�1 216�1
6�n�1
� 216 1 � �1�6�20
1 � �1�6� � 259.2
an � a1rn�1 � 216�1
6�n�1
a3 � a1r2 ⇒ 6 � a1�1
6�2
⇒ a1 � 63 � 216
1 � 6r ⇒ r �16
a4 � a3r
71. �7
i�1
2i�1 �1 � 27
1 � 2� 127 73.
� 3277
�7
n�1��4�n�1 �
1 � ��4�7
1 � ��4�
75. �4
n�0250�1.02�n � 250�1 � 1.025
1 � 1.02 � � 1301.01004
77. �10
i�1
10�3
5�i�1
� 24.849
72. �5
i�1
3i�1 �1 � 35
1 � 3� 121
74. �4
n�112��
12�
n�1
� 7.5
76. �5
n�0 400�1.08�n � 2934.3716 78. �
15
i�1 20�0.2�i�1 � 25
70.
�20
n�1 5�1
5�n�1
� 5�1 � �1�5�20
1 � 1�5 � 6.25
an � a1 rn�1 � 5�1
5�n�1
a1 � 5, r � 0.2
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.
Review Exercises for Chapter 8 765
86. 1. When
2. Assume that Then,
Thus, the formula holds for all positive integers n.
�k � 1
4��k � 1� � 3�.
��k � 1��k � 4�
4
�k2 � 5k � 4
4
�k�k � 3� � 2�k � 2�
4
1 �3
2� 2 �
5
2� . . . �
1
2�k � 1� �
1
2�k � 2� �
k
4�k � 3� �
1
2�k � 2�
1 �3
2� 2 �
5
2� . . . �
1
2�k � 1� �
k
4�k � 3�.
n � 1, 1 �1
4�1 � 3� � 1.
79. �4
1 � 7�8� 32�
�
i�14�7
8�i�1
� ��
i�04�7
8�i
80. �6
1 � 1�3� 9�
�
i�16�1
3�i�1
� ��
i�06�1
3�i
81. ��
k�1
4�2
3�k�1
�4
1 � 2�3� 12
83. (a)
(b) a5 � 120,000�0.7�5 � $20,168.40
at � 120,000�0.7�t
82. ��
k�1
1.3� 1
10�k�1
�1.3
1 � �1�10��
13
9
84. A � �48
i�175�1 �
0.0412 �i
� $3909.96
85. 1. When
2. Assume that Then,
Therefore, by mathematical induction, the formula is true for all positive integers n.
�k � 1
2�5�k � 1� � 1�.
�12
��5k � 4��k � 1��
�12
�5k2 � 9k � 4�
�k2
�5k � 1� � 5k � 2
� Sk � 5k � 2
Sk�1 � 2 � 7 � . . . � �5k � 3� � �5�k � 1� � 3�
Sk � 2 � 7 � . . . � �5k � 3� �k2
�5k � 1�.
n � 1, 2 �12
�5�1� � 1�.
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766 Chapter 8 Sequences, Series, and Probability
88. 1. When
2. Assume that using as the induction variable. Then,
Thus, the formula holds for all positive integers n.
�2ia � i�i � 1�d � 2a � 2id
2�
2a�i � 1� � id�i � 1�2
� �i � 1
2 ��2a � id�.
�i�1�1
k�0
�a � kd� �i
2�2a � �i � 1�d� � �a � id�
i�i�1
k�0
�a � kd� �i
2�2a � �i � 1�d�,
n � 1, a � 0 � d � a �1
2�2a � �1 � 1�d� � a.
87. 1. When
2. Assume that
Then,
Therefore, by mathematical induction, the formula is valid for all positive integer values of n.
�a�1 � rk � rk � rk�1�
1 � r� a�1 � rk�1�
1 � r.
Sk�1 � �k
i�0
ari � �k�1
i�0
ari � ark �a�1 � rk�
1 � r� ark
Sk � �k�1
i�0
ari �a�1 � rk�
1 � r.
n � 1, a � a�1 � r
1 � r�.
90. �10
n�1 n2 �
10�10 � 1��20 � 1�6
� 38589. �30
n�1n �
30�31�2
� 465
91.
� 4676 � 28 � 4648 �840�167�
30� 28
�7�8��15��3�7�2 � 3�7� � 1�
30�
7�8�2
�7
n�1�n4 � n� � �
7
n�1n4 � �
7
n�1n
92. �6
n�1 �n5 � n2� �
62�72��2 � 62 � 12 � 1�12
�6�7��2�6� � 1�
6� 12,201 � 91 � 12,110
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Miff
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.
Review Exercises for Chapter 8 767
95.
1 2 3 4 5
16 15 14 13 12
First differences:
Second difference: 0 0 0
Linear model: an � 17 � n
�1�1�1�1
an:
n:
a5 � 13 � 1 � 12
a4 � 14 � 1 � 13
a3 � a2 � 1 � 15 � 1 � 14
a2 � a1 � 1 � 16 � 1 � 15
a1 � f �1� � 16 96.
1 2 3 4 5
1 1 2 2 3
First differences: 0 1 0 1
Second differences: 1 1
Neither linear nor quadratic
�1
an:
n:
a5 � 5 � 2 � 3
a4 � 4 � 2 � 2
a3 � 3 � a2 � 2
a2 � 2 � a1 � 2 � 1 � 1
a1 � f �1� � 1
97. 10C8 � 45 99. �94� � 9C4 � 126
101. 4th number in 6th row is 6C3 � 20.
98. 12C5 � 792
100. �1412� � 14C12 � 91
94.
1 2 3 4 5
First differences:
Second differences:
Quadratic model
�2�2�2
�10�8�6�4
�31�21�13�7�3an:
n:
a5 � �21 � 2�5� � �21 � 10 � �31
a4 � �13 � 2�4� � �13 � 8 � �21
a3 � �7 � 2�3� � �7 � 6 � �13
a2 � a1 � 2�2� � �3 � 4 � �7
a1 � f �1� � �3
93.
1 2 3 4 5
5 10 15 20 25
First differences: 5 5 5 5
Second difference: 0 0 0
Linear model: an � 5n
an:
n:
a5 � a4 � 5 � 25
a4 � a3 � 5 � 20
a3 � a2 � 5 � 15
a2 � a1 � 5 � 5 � 5 � 10
a1 � f �1� � 5
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768 Chapter 8 Sequences, Series, and Probability
103. 5th number in 8th row is �84� � 8C4 � 70.102. The 8th entry in the 9th row is 36.
104. The 6th entry in the 10th row is 252. 105.
� x4 � 20x3 � 150x2 � 500x � 625
�x � 5�4 � x4 � 4x3�5� � 6x2�52� � 4x�53� � 54
107.
� a5 � 20a4b � 160a3b2 � 640a2b3 � 1280ab4 � 1024b5
�a � 4b�5 � a5 � 5a4�4b� � 10a3�4b�2 � 10a2�4b�3 � 5a�4b�4 � �4b�5
106. � y � 3�3 � y3 � 9y2 � 27y � 27
108. � 945x3y4 � 189x2y5 � 21xy6 � y7�3x � y�7 � 2187x7 � 5103x6y � 5103x5y2 � 2835x4y3
109.
� 1241 � 2520i
� 2401 � 2744i � 1176 � 224i � 16
�7 � 2i�4 � 74 � 4�7�3�2i� � 6�7�2�2i�2 � 4�7��2i�3 � �2i�4
111.
n�E� � 10
E � �1, 11�, �2, 10�, �3, 9�, �4, 8�, �5, 7�, �7, 5�, �8, 4�, �9, 3�, �10, 2�, �11, 1��
110.
� �236 � 115i
� 64 � 240i � 300 � 125i
�4 � 5i�3 � 43 � 3�4�2�5i� � 3�4��5i�2 � �5i�3
112. ways�2!��6!� � 1440 113. (a)
(b)
(c) �2��3��2��3� � 36 schedules
�2��3��6��3� � 108 schedules
�4��3��6��3� � 216 schedules
114. (a) possible calls
(b) calls
(c) calls10,000,000 � 2,000,000 � 8,000,000
2 � 106 � 2,000,000
107 � 10,000,000 115. 10C8 �10!2!8!
�10 � 9
2� 45
116. 8C6 �8!
2!6!�
8 � 72
� 28 117. 12P10 �12!2!
� 239,500,800 118. 6P4 �6!2!
� 360
119. 100C98 �100!2!98!
�100 � 99
2� 4950 120. 50C48 �
50!2!48!
�50 � 49
2� 1225
121. 1000P2 �1000!998!
� 1000�999� � 999,000 122. 500P2 �500!498!
� 500�499� � 249,500
123. permutations8!
2!2!2!1!1!�
8!8
� 7! � 5040 124. permutations9!
2!2!�
9!4
� 90,720
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Review Exercises for Chapter 8 769
125. ways10! � 3,628,800 126. ways�7C2 ��11C2 � � 21 � 55 � 1155
128. ways54C6 � 25,827,165
132. P�E� �n�E�n�S�
�1
5!�
1
120
134. �6
6��5
6��4
6��3
6��2
6��1
6� �6!
66�
720
46,656�
5
324
127. ways20C15 � 15,504
131.10
10�
1
9�
1
9133. (a)
(b)
(c)37
500� 0.074
400
500� 0.8
208
500� 0.416
135. P�2 pairs� ��13C2��4C2��4C2��44C1�
�52C5�� 0.0475
137. True
�n � 2�!n!
��n � 2��n � 1�n!
n!� �n � 2��n � 1�
129.
n � 3
�n � 1�! � 4 � n!
�n � 1�!�n � 1�! � 4 �
n!�n � 1�!
n�1P2 � 4 � nP1 130.
n � 7
8n! � �n � 1�!
8n!
�n � 2�! ��n � 1�!�n � 2�!
8 � nP2 � n�1P3
136.
P�not club� � 1 �14
�34
P�club� �1352
�14
138. True 139. Answers will vary. See pages 526 and 535.
140. They differ by a minus sign.
(a)(Odd-numbered terms are negative.)
(b)(Even-numbered terms are negative.)1, �1
2, 13, . . .
�1, 12, �13, . . .
141. (a) Arithmetic-linear model
(b) Geometric model
142.
S10 � 1490 � 810 � 440 � 2740
S9 � 810 � 440 � 240 � 1490
S8 � 440 � 240 � 130 � 810
S7 � 240 � 130 � 70 � 440
S6 � 130 � 70 � 40 � 240
144. When an � an�1�r� < an�1.0 < r < 1, 145. If n is even, the expansion are the same. If n isodd, the expansion of is the negativeof that of �x � y�n.
��x � y�n
146. In the closed interval �0, 1�.
143. Answers will vary. See page 528. To define asequence recursively, you need to be given one or more of the first few terms. All other terms are defined using previous terms.
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770 Chapter 8 Sequences, Series, and Probability
Chapter 8 Practice Test
1. Write out the first five terms of the sequence an �2n
�n � 2�!.
2. Write an expression for the nth term of the sequence �43, 59, 6
27, 781, 8
243, . . .�.
3. Find the sum �6
i�1
�2i � 1�.
4. Write out the first five terms of the arithmetic sequence where and d � �2.a1 � 23
20. A manufacturer has determined that for every 1000 units it produces, 3 will be faulty. What is the probability that an order of 50 units will have one or more faulty units?
5. Find for the arithmetic sequence with a1 � 12, d � 3, and n � 50.a50
6. Find the sum of the first 200 positive integers.
7. Write out the first five terms of the geometric sequence with a1 � 7 and r � 2.
8. Evaluate �9
n�0
6�2
3�n
. 9. Evaluate ��
n�0
�0.03�n.
10. Use mathematical induction to prove that 1 � 2 � 3 � 4 � . . . � n �n�n � 1�
2.
11. Use mathematical induction to prove that n! > 2n, n ≥ 4.
12. Evaluate Verify with a graphing utility.13C4.
13. Expand �x � 3�5.
14. Find the term involving x7 in �x � 2�12.
15. Evaluate 30P4.
16. How many ways can six people sit at a table with six chairs?
17. Twelve cars run in a race. How many different ways can they come in first, second, and third place? (Assume that there are no ties.)
18. Two six-sided dice are tossed. Find the probability that the total of the two dice is less than 5.
19. Two cards are selected at random from a deck of 52 playing cards without replacement. Find the probability that the first card is a King and the second card is a black ten.
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