Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12: Sequences; Induction; the Binomial Theorem
85. a. Find the 10th term of the geometric
sequence:
a1
= 2, r = 0.9, n=
10
10−1 9
a = 2(0.9) = 2(0.9) = 0.775 feet
10
b. Find n when a n < 1:
n−1
2(0.9) < 1
n−1
( 0.9)
< 0.5
( ) ( )
log ( 0.5)
n − 1 >
log ( 0.9)
log ( 0.5)
n
log ( 0.9)
( n − 1)log 0.9 < log 0.5
> + 1≈7.58
On the 8th swing the arc is less than 1 foot.
c. Find the sum of the first 15 swings:
15
⎛
15
1 − (0.9) ⎞ ⎛1−
( 0.9)
⎞
S15
= 2 2
⎜ 1 0.9 ⎟
=
⎝ − ⎠
⎜ 0.1 ⎟
⎝ ⎠
15
= 20 1− 0.9 = 15.88 feet
( ( ) )
d. Find the infinite sum of the geometric series:
2 2
S ∞ = = = 20 feet
1−
0.9 0.1
86. a. Find the 3rd term of the geometric
sequence:
a1
= 24, r = 0.8, n = 3
3−1 2
a = 24(0.8) = 24(0.8) = 15.36 feet
3
b. The height after the n th bounce is:
n−1
−1
n
a = 24(0.8) = 24 0.8 0.8
n
( )
n
= 30 0.8 ft
c. Find n when a n < 0.5 :
n−1
24(0.8) < 0.5
( ) ( )
n−1
( 0.8)
< 0.020833
( ) ( )
log ( 0.020833)
n − 1 >
log ( 0.8)
log ( 0.020833)
n
log ( 0.8)
( n − 1)log 0.8 < log 0.020833
> + 1≈18.35
On the 19th bounce the height is less than
0.5 feet.
d. Find the infinite sum of the geometric
series:
24 24
S ∞ = = = 120 feet on the upward
1−
0.8 0.2
bounce.
For the downward motion of the ball:
30 30
S ∞ = = = 150 feet
1−
0.8 0.2
The total distance the ball travels is
120 + 150 = 270 feet.
87. This is a geometric sequence with
a1 = 1, r = 2, n = 64 .
Find the sum of the geometric series:
⎛
64 64
1−2 ⎞ 1−2
64
S64
= 1 ⎜
2 1
1 2 ⎟
= = −
⎝ − ⎠ −1
19
= 1.845×
10 grains
88. This is an infinite geometric series with
a 1 1
1 = , r = .
4 4
Find the sum of the infinite geometric series:
1 1
( ) ( ) 1
S
4 4
∞ = = =
1 3
( 1−
) ( )
3
4 4
Therefore, 1 3
of the square is eventually shaded.
89. The common ratio, r = 0.90 < 1. The sum is:
1 1
S = = = 10 .
1−
0.9 0.10
The multiplier is 10.
90. The common ratio, r = 0.95 < 1. The sum is:
1 1
S = = = 20 .
1−
0.95 0.05
The multiplier is 20.
91. This is an infinite geometric series with
1.03
a = 4, and r = .
1.09
Find the sum:
4
Price = $72.67
1.03
≈ per share.
⎛ ⎞
⎜1−
⎟
⎝ 1.09 ⎠
1260
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