82
CHAPTER 8 Sequences, 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 © Houghton Mifflin Company. All rights reserved.

CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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Page 1: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 2: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

C H A P T E R 8Sequences, Series, and Probability

Section 8.1 Sequences and Series

690

â–  Given the general th term in a sequence, you should be able to find, or list, some of the terms.

â–  You should be able to find an expression for the th term of a sequence.

â–  You should be able to use and evaluate factorials.

â–  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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Page 3: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 4: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 5: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 6: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 7: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 8: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 9: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 10: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 11: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 12: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 13: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 14: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 15: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 16: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 17: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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

â–  You should be able to recognize an arithmetic sequence, find its common difference, and find its th term.

â–  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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Page 18: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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

â–  You should be able to identify a geometric sequence, find its common ratio, and find the th term.

â–  You should be able to find the th partial sum of a geometric sequence with common ratio usingthe formula.

â–  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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Page 27: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 28: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 29: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 30: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 31: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 32: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 33: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 34: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 35: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 36: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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

â–  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

â–  You should be able to verify (by induction) the formulas for the sums of powers of integers and be able touse these formulas.

â–  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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Page 37: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 38: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 39: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 40: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 41: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 42: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 43: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 44: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 46: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 47: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 50: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

Section 8.5 The Binomial Theorem

738 Chapter 8 Sequences, Series, and Probability

â–  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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Page 59: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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�©

Hou

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n M

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Com

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. All

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Page 60: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

748 Chapter 8 Sequences, Series, and Probability

Section 8.6 Counting Principles

â–  You should know The Fundamental Counting Principle.

â–  is the number of permutations of n elements taken r at a time.

â–  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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Page 61: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 62: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 63: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 64: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 65: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

Section 8.7 Probability 753

You should know the following basic principles of probability.

â–  If an event E has equally likely outcomes and its sample space has equally likely outcomes, then theprobability of event E is

where

â–  If A and B are mutually exclusive events, then

If A and B are not mutually exclusive events, then

â–  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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Page 66: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 67: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 68: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 69: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 70: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 71: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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

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Page 72: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 73: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 74: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 75: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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

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Page 76: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 77: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 78: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 79: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 80: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 81: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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Page 82: CHAPTER 8 Sequences, Series, and Probabilitycrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch8.pdfCHAPTER 8 Sequences, Series, and Probability Section 8.1 Sequences and Series 690

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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