10-3 Geometric Sequences and Series
Determine the common ratio, and find the next three terms of each geometric sequence.
2. SOLUTION: First, find the common ratio.
The common ratio is
. Multiply the third term by
to find the fourth term, and so on.
Therefore, the next three terms are –
,
, and –
.
ANSWER: 4. 8, 20, 50, …
SOLUTION: First, find the common ratio.
20 ÷ 8 or 2.5
50 ÷ 20 or 2.5
The common ratio is 2.5. Multiply the third term by 2.5 to find the fourth term, and so on.
50(2.5) = 125
125(2.5) = 312.5
312.5(2.5) = 781.25
Therefore, the next three terms are 125, 312.5, and 781.25.
ANSWER: 2.5; 125, 312.5, 781.25
Write an explicit formula and a recursive formula for finding the nth term of each geometric sequence.
10. 36, 12, 4, …
SOLUTION: First, find the common ratio.
12 ÷ 36 =
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4 ÷ 12 =
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Therefore, the next three terms are 125, 312.5, and 781.25.
10-3 ANSWER: Geometric Sequences and Series
2.5; 125, 312.5, 781.25
Write an explicit formula and a recursive formula for finding the nth term of each geometric sequence.
10. 36, 12, 4, …
SOLUTION: First, find the common ratio.
12 ÷ 36 =
4 ÷ 12 =
For an explicit formula, substitute a 1 = 36 and r =
in the nth term formula.
For a recursive formula, state the first term a 1. Then indicate that the next term is the product of the first term a n –
1
and r.
a 1 = 36, a n =
ANSWER: a n = 36
; a 1 = 36, a n =
an – 1
12. −2, 10, −50, …
SOLUTION: First, find the common ratio.
10 ÷ –2 = –5
–50 ÷ 10 = –5
For an explicit formula, substitute a 1 = –2 and r = –5 in the nth term formula.
For a recursive formula, state the first term a 1. Then indicate that the next term is the product of the first term a n –
1
and r.
a 1 = –2, a n =
ANSWER: n−1
a n = −2(−5)
16. 15, 5,
,…
; a 1 = −2, a n = −5a n − 1
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SOLUTION: Page 2
a 1 = –2, a n =
ANSWER: n − 1Sequences and Series
10-3 aGeometric
= −2(−5)
; a = −2, a = −5a
n
1
16. 15, 5,
n
n−1
,…
SOLUTION: First, find the common ratio.
5 ÷ 15 =
÷ 5 =
For an explicit formula, substitute a 1 = 15 and r =
in the nth term formula.
For a recursive formula, state the first term a 1. Then indicate that the next term is the product of the first term a n –
1
and r.
a 1 = 15, a n =
ANSWER: a n = 15
; a 1 = 15, a n =
an − 1
Find the specified term for each geometric sequence or sequence with the given characteristics.
20. a 9 for 60, 30, 15, …
SOLUTION: First, find the common ratio.
30 ÷ 60 =
15 ÷ 30 =
Use the formula for the nth term of a geometric sequence to find a 9.
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ANSWER: Page 3
ANSWER: 10-3 aGeometric
;Sequences
a 1 = 15, a n = and
a n −Series
n = 15
1
Find the specified term for each geometric sequence or sequence with the given characteristics.
20. a 9 for 60, 30, 15, …
SOLUTION: First, find the common ratio.
30 ÷ 60 =
15 ÷ 30 =
Use the formula for the nth term of a geometric sequence to find a 9.
ANSWER: 26. a 6 if a 1 = 16,807 and r =
SOLUTION: Use the formula for the nth term of a geometric sequence to find a 6.
ANSWER: 243
Find the sum of each geometric series described.
40. first six terms of 3 + 9 + 27 + …
SOLUTION: First,
find- Powered
the common
ratio.
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Manual
by Cognero
9÷3=3
27 ÷ 9 = 3
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10-3 ANSWER: Geometric Sequences and Series
243
Find the sum of each geometric series described.
40. first six terms of 3 + 9 + 27 + …
SOLUTION: First, find the common ratio.
9÷3=3
27 ÷ 9 = 3
The common ratio is 3. Use Formula 1 for the sum of a finite geometric series.
ANSWER: 1092
43. first n terms of a 1 = 4, a n = 2000, r = −3
SOLUTION: Use Formula 2 for the nth partial sum of a geometric series.
ANSWER: 1501
46. first n terms of a 1 = −8, a n = −256, r = 2
SOLUTION: Use Formula 2 for the nth partial sum of a geometric series.
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10-3 ANSWER: Geometric Sequences and Series
1501
46. first n terms of a 1 = −8, a n = −256, r = 2
SOLUTION: Use Formula 2 for the nth partial sum of a geometric series.
ANSWER: −504
Find each sum.
48. SOLUTION: Find n, a 1, and r.
Substitute n = 6, a 1 = 5, and r = 2 into the formula for the sum of a finite geometric series.
ANSWER: 315
If possible, find the sum of each infinite geometric series.
58. 18 + (–27) + 40.5 + ...
SOLUTION: First, find the common ratio.
–27Manual
÷ 18-=Powered
–1.5 by Cognero
eSolutions
40.5 ÷ –27 = –1.5
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ANSWER: 10-3 315
Geometric Sequences and Series
If possible, find the sum of each infinite geometric series.
58. 18 + (–27) + 40.5 + ...
SOLUTION: First, find the common ratio.
–27 ÷ 18 = –1.5
40.5 ÷ –27 = –1.5
> 1. Therefore, this infinite geometric series has no sum.
The common ratio r is
ANSWER: does not exist
59. 12 + (–7.2) + 4.32 + ...
SOLUTION: First, find the common ratio.
–7.2 ÷ 12 = –0.6
4.32 ÷ –7.2 = –0.6
The common ratio r is
< 1. Therefore, this infinite geometric series has a sum. Use the formula for the sum of an infinite geometric series.
Therefore, the sum of the series is 7.5.
ANSWER: 7.5
60. SOLUTION: The common ratio is
< 1. Therefore, this infinite geometric series has a sum. Find a 1.
Use the formula for the sum of an infinite geometric series to find the sum.
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Therefore, the sum of the series is 7.5.
10-3 ANSWER: Geometric Sequences and Series
7.5
60. SOLUTION: The common ratio is
< 1. Therefore, this infinite geometric series has a sum. Find a 1.
Use the formula for the sum of an infinite geometric series to find the sum.
Therefore, the sum of the series is
.
ANSWER: Write each geometric series in sigma notation.
98. 9 + 18 + 36 + … + 1152
SOLUTION: Find the common ratio.
18 ÷ 9 = 2
36 ÷ 18 = 2
Next, determine the upper bound.
a 4 = 36(2) = 72
a 5 = 72(2) = 144
a 6 = 144(2) = 288
a 7 = 288(2) = 576
a 8 = 576(2) = 1152
Write an explicit formula for the sequence.
Therefore, in sigma notation the series 9 + 18 + 36 + … + 1152 can be written as
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ANSWER: 10-3 Geometric Sequences and Series
Write each geometric series in sigma notation.
98. 9 + 18 + 36 + … + 1152
SOLUTION: Find the common ratio.
18 ÷ 9 = 2
36 ÷ 18 = 2
Next, determine the upper bound.
a 4 = 36(2) = 72
a 5 = 72(2) = 144
a 6 = 144(2) = 288
a 7 = 288(2) = 576
a 8 = 576(2) = 1152
Write an explicit formula for the sequence.
Therefore, in sigma notation the series 9 + 18 + 36 + … + 1152 can be written as
ANSWER: 100. SOLUTION: Find the common ratio.
–
÷
= –2
÷ –
= –2
Next, determine the upper bound.
a4 =
a 5 = –1(–2) = 2
a 6 = 2(–2) = –4
a 7 = –4(–2) = 8
Write an explicit formula for the sequence.
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ANSWER: 10-3 Geometric Sequences and Series
100. SOLUTION: Find the common ratio.
–
÷
= –2
÷ –
= –2
Next, determine the upper bound.
a4 =
a 5 = –1(–2) = 2
a 6 = 2(–2) = –4
a 7 = –4(–2) = 8
Write an explicit formula for the sequence.
Therefore, in sigma notation the series
can be written as ANSWER: eSolutions Manual - Powered by Cognero
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