10-4 Infinite Geometric Series
Determine whether each infinite geometric series is convergent or divergent.
1. 16 – 8 + 4 – …
SOLUTION: Find the value of r.
Since
, the series is convergent.
3. 0.5 + 0.7 + 0.98 + …
SOLUTION: Find the value of r.
Since
, the series is divergent.
Find the sum of each infinite series, if it exists.
5. 440 + 220 + 110 + …
SOLUTION: Find the value of r.
Since
, the series is convergent.
Find the sum.
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440 + 220 + 110 + … = 880
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, the series
is divergent.
10-4Since
Infinite Geometric
Series
Find the sum of each infinite series, if it exists.
5. 440 + 220 + 110 + …
SOLUTION: Find the value of r.
Since
, the series is convergent.
Find the sum.
440 + 220 + 110 + … = 880
7. SOLUTION: Find the value of r.
Since
, the series is diverges and the sum does not exist.
9. CCSS SENSE-MAKING A certain drug has a half-life of 8 hours after it is administered to a patient. What
percent of the drug is still in the patient’s system after 24 hours?
SOLUTION: Given a 1 = 100% or 1 and r =
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and n = 4.
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Since
, the series is diverges and the sum does not exist.
10-4 Infinite Geometric Series
9. CCSS SENSE-MAKING A certain drug has a half-life of 8 hours after it is administered to a patient. What
percent of the drug is still in the patient’s system after 24 hours?
SOLUTION: Given a 1 = 100% or 1 and r =
and n = 4.
After 24 hours, 12.5% of the drug is still in the patient’s system.
Find the sum of each infinite series, if it exists.
11. SOLUTION: Since
, the series is convergent.
13. SOLUTION: Since
, the series is convergent.
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10-4 Infinite Geometric Series
13. SOLUTION: Since
, the series is convergent.
Write each repeating decimal as a fraction.
15. SOLUTION: Find the value of r.
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10-4 Infinite Geometric Series
Write each repeating decimal as a fraction.
15. SOLUTION: Find the value of r.
Determine whether each infinite geometric series is convergent or divergent.
17. 480 + 360 + 270 + …
SOLUTION: Find the value of r.
Since
, the series is convergent.
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SOLUTION: Find the value of r.
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Since
, the series is convergent.
10-4 Infinite Geometric Series
19. SOLUTION: Find the value of r.
Since
, the series is divergent.
21. 0.008 + 0.08 + 0.8 + …
SOLUTION: Find the value of r.
Since
, the series is divergent.
Find the sum of each infinite series, if it exists.
23. –3 – 4.2 – 5.88 – …
SOLUTION: Find the value of r.
Since
, the series is diverges and the sum does not exists.
25. SOLUTION: Find the value of r.
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Since
, the series is diverges and the sum does not exists.
10-4 Infinite Geometric Series
25. SOLUTION: Find the value of r.
Since
, the series is convergent.
Find the sum.
27. 32 + 40 + 50 + …
SOLUTION: Find the value of r.
Since
, the series is diverges and the sum does not exists.
Find the sum of each infinite series, if it exists.
29. SOLUTION: eSolutions Manual - Powered by Cognero
Since
, the series is diverges and the sum does not exists.
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, the series
is diverges and the sum does not exists.
10-4Since
Infinite Geometric
Series
Find the sum of each infinite series, if it exists.
29. SOLUTION: Since
, the series is diverges and the sum does not exists.
31. SOLUTION: Since
, the series is convergent.
33. SOLUTION: Since
, the series is convergent.
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10-4 Infinite Geometric Series
33. SOLUTION: Since
, the series is convergent.
Write each repeating decimal as a fraction.
35. SOLUTION: The number
can be written as .
Find the value of r.
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Therefore:
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10-4 Infinite Geometric Series
Write each repeating decimal as a fraction.
35. SOLUTION: The number
can be written as .
Find the value of r.
Therefore:
37. SOLUTION: The number
can be written as .
Find the value of r.
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10-4 Infinite Geometric Series
37. SOLUTION: The number
can be written as .
Find the value of r.
Therefore:
39. SOLUTION: The number
can be written as .
Find the value of r.
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10-4 Infinite Geometric Series
39. SOLUTION: The number
can be written as .
Find the value of r.
Therefore:
41. FANS A fan is running at 10 revolutions per second. After it is turned off, its speed decreases at a rate of 75% per
second. Determine the number of revolutions completed by the fan after it is turned off.
SOLUTION: Given a 1 = 10 and r = 100% – 75% or 0.25.
Find the sum.
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10-4 Infinite Geometric Series
41. FANS A fan is running at 10 revolutions per second. After it is turned off, its speed decreases at a rate of 75% per
second. Determine the number of revolutions completed by the fan after it is turned off.
SOLUTION: Given a 1 = 10 and r = 100% – 75% or 0.25.
Find the sum.
The fan completed
revolutions after it is turned off.
43. RECHARGEABLE BATTERIES A certain rechargeable battery is advertised to recharge back to 99.9% of its
previous capacity with every charge. If its initial capacity is 8 hours of life, how many total hours should the battery
last?
SOLUTION: Given a 1 = 8 and r = 99.9% or 0.999
Find the sum.
Find the sum of each infinite series, if it exists.
45. SOLUTION: Find the value of r.
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10-4 Infinite Geometric Series
Find the sum of each infinite series, if it exists.
45. SOLUTION: Find the value of r.
Since
, the series is convergent.
Find the sum.
47. SOLUTION: Find the value of r.
Since
, the series is diverges and the sum does not exists.
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49. Page 14
10-4 Infinite Geometric Series
47. SOLUTION: Find the value of r.
Since
, the series is diverges and the sum does not exists.
49. SOLUTION: Find the value of r.
Since
, the series is convergent.
Find the sum.
51. PHYSICS In a physics experiment, a steel ball on a flat track is accelerated, and then allowed to roll freely. After
the first minute, the ball has rolled 120 feet. Each minute the ball travels only 40% as far as it did during the
preceding minute. How far does the ball travel?
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SOLUTION: Page 15
10-4 Infinite Geometric Series
51. PHYSICS In a physics experiment, a steel ball on a flat track is accelerated, and then allowed to roll freely. After
the first minute, the ball has rolled 120 feet. Each minute the ball travels only 40% as far as it did during the
preceding minute. How far does the ball travel?
SOLUTION: Given a 1 = 120 and r = 40% or 0.4.
Find the sum.
The ball travels 200 ft.
53. TOYS If a rubber ball can bounce back to 95% of its original height, what is the total vertical distance that it will
travel if it is dropped from an elevation of 30 feet?
SOLUTION: Distance traveled by the rubber ball in downward direction.
Given, a 1 = 30 and r = 95% or 0.95.
Find the sum.
Distance traveled by the rubber ball in upward direction.
Given, a 1 = 28.5 and r = 95% or 0.95.
Find the sum.
The Manual
total distance
by
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by Cognero
the rubber ball is 600 + 570 or 1170 ft.
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ball travels
200 ft.Series
10-4The
Infinite
Geometric
53. TOYS If a rubber ball can bounce back to 95% of its original height, what is the total vertical distance that it will
travel if it is dropped from an elevation of 30 feet?
SOLUTION: Distance traveled by the rubber ball in downward direction.
Given, a 1 = 30 and r = 95% or 0.95.
Find the sum.
Distance traveled by the rubber ball in upward direction.
Given, a 1 = 28.5 and r = 95% or 0.95.
Find the sum.
The total distance traveled by the rubber ball is 600 + 570 or 1170 ft.
55. ECONOMICS A state government decides to stimulate its economy by giving $500 to every adult. The government
assumes that everyone who receives the money will spend 80% on consumer goods and that the producers of these
goods will in turn spend 80% on consumer goods. How much money is generated for the economy for every $500
that the government provides?
SOLUTION: Here, a 1 = 500 and r = 80% or 0.8.
Find the sum.
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Match each graph with its corresponding description.
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total distance
traveled
by the rubber ball is 600 + 570 or 1170 ft.
10-4The
Infinite
Geometric
Series
55. ECONOMICS A state government decides to stimulate its economy by giving $500 to every adult. The government
assumes that everyone who receives the money will spend 80% on consumer goods and that the producers of these
goods will in turn spend 80% on consumer goods. How much money is generated for the economy for every $500
that the government provides?
SOLUTION: Here, a 1 = 500 and r = 80% or 0.8.
Find the sum.
Match each graph with its corresponding description.
57. a. converging geometric series
b. diverging geometric series
c. converging arithmetic series
d. diverging arithmetic series
SOLUTION: The graph is diverging geometric series. Therefore, option b is the correct answer.
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a. converging geometric series
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SOLUTION: The graph is diverging geometric series. Therefore, option b is the correct answer.
10-4 Infinite Geometric Series
59. a. converging geometric series
b. diverging geometric series
c. converging arithmetic series
d. diverging arithmetic series
SOLUTION: The graph is converging geometric series. Therefore, option a is the correct answer.
61. PROOF Derive the formula for the sum of an infinite geometric series.
SOLUTION: Sample answer:
The sum of a geometric series is
.
For an infinite series with | r | < 1,
.
Thus,
63. REASONING When does an infinite geometric series have a sum, and when does it not have a sum? Explain your
reasoning.
SOLUTION: Sample answer: An infinite geometric series has a sum when the common ratio has an absolute value less than 1.
When this occurs, the terms will approach 0 as n approaches infinity. With the future terms almost 0, the sum of the
series will approach a limit. When the common ratio is 1 or greater, the terms will keep increasing and approach
infinity as n approaches infinity and the sum of the series will have no limit.
65. OPEN ENDED Write an infinite series with a sum that converges to 9.
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Sample answer:
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Sample answer: An infinite geometric series has a sum when the common ratio has an absolute value less than 1.
When this occurs, the terms will approach 0 as n approaches infinity. With the future terms almost 0, the sum of the
series will approach a limit. When the common ratio is 1 or greater, the terms will keep increasing and approach
as Geometric
n approachesSeries
infinity and the sum of the series will have no limit.
10-4infinity
Infinite
65. OPEN ENDED Write an infinite series with a sum that converges to 9.
SOLUTION: Sample answer:
67. WRITING IN MATH Explain why an arithmetic series is always divergent.
SOLUTION: An arithmetic series has a common difference, so each term will eventually become more positive or more negative,
but never approach 0. With the terms not approaching 0, the sum will never reach a limit and the series cannot
converge.
69. Adelina, Michelle, Masao, and Brandon each simplified the same expression at the board. Each student’s work is
shown below. The teacher said that while two of them had a correct answer, only one of them had arrived at the
correct conclusion using correct steps.
Adelina’s work
Masao’s work
Michelle’s work
Brandon’s work
Which is a completely accurate simplification?
F Adelina’s work
G Michelle’s work
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H Masao’s work
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SOLUTION: An arithmetic series has a common difference, so each term will eventually become more positive or more negative,
but never approach 0. With the terms not approaching 0, the sum will never reach a limit and the series cannot
10-4converge.
Infinite Geometric Series
69. Adelina, Michelle, Masao, and Brandon each simplified the same expression at the board. Each student’s work is
shown below. The teacher said that while two of them had a correct answer, only one of them had arrived at the
correct conclusion using correct steps.
Adelina’s work
Masao’s work
Michelle’s work
Brandon’s work
Which is a completely accurate simplification?
F Adelina’s work
G Michelle’s work
H Masao’s work
J Brandon’s work
SOLUTION: Masao’s work is a completely accurate simplification.
Option H is the correct answer.
71. GEOMETRY The radius of a large sphere was multiplied by a factor of
to produce a smaller sphere.
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SOLUTION: Masao’s work is a completely accurate simplification.
H Geometric
is the correctSeries
answer.
10-4Option
Infinite
71. GEOMETRY The radius of a large sphere was multiplied by a factor of
to produce a smaller sphere.
How does the volume of the smaller sphere compare to the volume of the larger sphere?
A The volume of the smaller sphere is
as large.
B The volume of the smaller sphere is
as large.
C The volume of the smaller sphere is
as large.
D The volume of the smaller sphere is
as large.
SOLUTION: The volume of the smaller sphere is
as large.
Option C is the correct answer.
73. CLUBS A quilting club consists of 9 members. Every week, each member must bring one completed quilt square.
a. Find the first eight terms of the sequence that describes the total number of squares that have been made after
each meeting.
b. One particular quilt measures 72 inches by 84 inches and is being designed with 4-inch squares. After how many
meetings will the quilt be complete?
SOLUTION: a. 9, 18, 27, 36, 45, 54, 63, 72
b. The number of squares required is
.
That is a n = 378.
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Given a 1 = 9 and d = 9.
Find n.
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The volume of the smaller sphere is
as large.
C Geometric
is the correctSeries
answer.
10-4Option
Infinite
73. CLUBS A quilting club consists of 9 members. Every week, each member must bring one completed quilt square.
a. Find the first eight terms of the sequence that describes the total number of squares that have been made after
each meeting.
b. One particular quilt measures 72 inches by 84 inches and is being designed with 4-inch squares. After how many
meetings will the quilt be complete?
SOLUTION: a. 9, 18, 27, 36, 45, 54, 63, 72
b. The number of squares required is
.
That is a n = 378.
Given a 1 = 9 and d = 9.
Find n.
Find each function value.
2
75. g(x) = x – x, g(4)
SOLUTION: Substitute 4 for x and evaluate.
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