College Algebra 9th txtbk

SECTION 8–3 Arithmetic and Geometric Sequences 529
19. a 1 81, r 1 3; a 10 ?
20. a 1 64, r 1 2; a 13 ?
21. a 1 3, a 7 2,187, r 3; S 7 ?
22. a 1 1, a 7 729, r 3; S 7 ?
Let a 1 , a 2 , a 3 , . . . , a n ,... be an arithmetic sequence. In Problems
23–30, find the indicated quantities.
23. a 1 3, a 20 117; d ?, a 101 ?
24. a 1 7, a 8 28; d ?, a 25 ?
25. a 1 12, a 40 22; S 40 ?
26. a 1 24, a 24 28; S 24 ?
27. a 1 1 3, a 2 1 2; a 11 ?, S 11 ?
28. a 1 1 6, a 2 1 4; a 19 ?, S 19 ?
29. a 3 13, a 10 55; a 1 ?
30. a 9 12, a 13 3; a 1 ?
Let a 1 , a 2 , a 3 , . . . , a n ,... be a geometric sequence. Find each of
the indicated quantities in Problems 31–42.
31. a 1 8, a 2 2; r ?
32. a 1 6, a 2 2; r ?
33. a 1 120, a 4 15; r ?
34. a 1 12, a 6 8; r ?
35. a 1 9, r 2 3; S 10 ?
36. a 1 3, r 5; S 9 ?
37. a 1 1, a 8 2,187; S 8 ?
38. a 1 1 2, a 12 1,024; S 12 ?
39. a 3 72, a 6 243; a 1 ?
40. a 4 8, a 5 6; a 1 ?
41. a 1 1, a 4 1; a 100 ?
42. a 1 1, a 8 1; a 99 ?
51
43. S 51 a (3k 3) ?
k1
40
44. S 40 a (2k 3) ?
k1
7
45. S 7 a (3) k1 ?
k1
7
46. S 7 a 3 k ?
k1
47. Find the sum of all the even integers between 21 and 135.
48. Find the sum of all the odd integers between 100 and 500.
49. Show that the sum of the first n odd natural numbers is n 2 ,
using appropriate formulas from Section 8-3.
50. Show that the sum of the first n even natural numbers is
n n 2 , using appropriate formulas from Section 8-3.
In Problems 51–60, find the sum of each infinite geometric series
that has a sum.
51. 2 1 2 1 8 . . .
52. 6 2 2 3 . . .
53. 3 1 1 3 . . .
54. 1 4 3 16
9 . . .
55. 1 0.1 0.01 ...
56. 10 2 0.4 ...
57. 1 1 2 1 4 . . .
58. 6 4 8 3 . . .
59. 1 1 1 ...
60. 100 80 64 ...
In Problems 61–66, represent each repeating decimal as the quotient
of two integers.
61. 0.7 0.7777 . . .
62. 0.5 0.5555 . . .
63. 0.54 0.545 454 . . . 64. 0.27 0.272 727 . . .
65. 3.216 3.216 216 216 . . .
66. 5.63 5.636 363 . . .
67. Prove, using mathematical induction, that if {a n } is an arithmetic
sequence, then
a n a 1 (n 1)d
for every n 7 1
68. Prove, using mathematical induction, that if {a n } is an arithmetic
sequence, then
S n n 2 [2a 1 (n 1)d]
69. If in a given sequence, a 1 2 and a n 3a n1 , n 7 1, find
a n in terms of n.
n
70. For the sequence in Problem 69, find S n a a k in terms
k1
of n.
71. Prove, using mathematical induction, that if {a n } is a geometric
sequence, then
n1
a n a 1 r
72. Prove, using mathematical induction, that if {a n } is a geometric
sequence, then
S n a n
1 a 1 r
1 r
n N
n N, r 1
73. Is there an arithmetic sequence that is also geometric?
Explain.
74. Is there an infinite geometric sequence with a 1 1 that has
1
sum equal to 2? Explain.