Chapter 10
1.
Find the sum of
2 3 7
4 4 4 4+ + ++
.
If it does not exist, enter “DNE”.
2.
Find the sum of
23
90 90(1.04) 90(1.04) 90(1.04)+ + + +
.
If it does not exist, enter “DNE”.
Ans:
section: 10.1
3.
Find the sum of
600 300 150 75+ + + +
.
If it does not exist, enter “DNE”.
Ans:
1200
section: 10.1
4.
Find the sum of
1 1 1
14 16 64
− + − +
.
If it does not exist, enter “DNE”.
section: 10.1
5.
Each quarter, $1000 is deposited into an account earning 1.2% interest per quarter,
compounded quarterly. How much money is in the account right before the 11th
deposit? Round to the nearest dollar.
Ans:
$10,684
section: 10.1
Ans:
21,844
section: 10.1
Chapter 10
6.
Consider the sum
23
10 10(0.5) 10(0.5) 10(0.5)S= + + + +
.
A. Calculate the partial sum
10
S
. Round to 4 decimal places.
B. Calculate the infinite sum S. Round to 4 decimal places.
Part A:
19.9805
Part B:
20.0000
Learning Objectives: Compute sum of finite geometric series.; Compute sum of infinite
geometric series. difficulty: easy section: 10.1
7.
Each week, a patient is given a 40 mg dose of an experimental vaccine, and 25% of the
vaccine remains in the body after one week. How many milligrams of the vaccine are
in the body right after the 15th dose? Round to 2 decimal places.
Ans:
53.33
Learning Objectives: Compute sum of finite geometric series. difficulty: medium
section: 10.1
8.
Does the infinite series
3/2 2
6 6 6 6
63 3 3
3
+ + + + +
converge or diverge?
Ans:
converges
Learning Objectives: Compute sum of infinite geometric series. difficulty: easy
section: 10.1
9.
Find
5
S
for the series
3/2 2
4 4 4 4
45 5 5
5
+ + + + +
Round to 3 decimal places.
Ans:
7.107
Learning Objectives: Compute sum of finite geometric series. difficulty: medium
section: 10.1
10.
Find the sum of the series
16
5
6
5
n
n=
. Round to 2 decimal places.
Ans:
98.49
Learning Objectives: Compute sum of finite geometric series. difficulty: medium
section: 10.1
Chapter 10
11.
If the sum of the series
3 9 27 81
a a a a
Sa= + + + +
is
2
3
11
3
23
−
, what is a?
Ans:
3
Learning Objectives: Compute sum of finite geometric series. difficulty: hard
section: 10.1
12.
A ball is dropped from a height of 10 feet and bounces. Each bounce is 2/3 the height
of the bounce before. Find an expression for the height to which the ball rises after it
hits the floor for the nth time, and use it to find the total vertical distance the ball has
traveled when it hits the floor for the 4th time. Round to 2 decimal places.
Ans:
38.15
Learning Objectives: Compute sum of finite geometric series. difficulty: medium
section: 10.1
13.
A tennis ball is dropped from a height of 50 feet and bounces. Each bounce is 1/2 the
height of the bounce before. A superball is has a bounce of 3/4 the height of the
bounce before and is dropped from a height of 30 feet. Which ball bounces a greater
total vertical distance?
A)
the superball
B)
the tennis ball
Ans: A Learning Objectives: Compute sum of finite geometric series.
difficulty: medium section: 10.1
14.
Is the following a geometric series? Answer “Yes” or “No”.
2 + 4 + 8 + 16 + 32 + …
Ans:
Yes
Learning Objectives: Determine whether or not a series is geometric.
difficulty: easy section: 10.1
15.
Is the following a geometric series? Answer “Yes” or “No”.
13 13 13 13 13 …
5 15 45 135 405
+ + + + +
Ans:
Yes
Learning Objectives: Determine whether or not a series is geometric.
difficulty: easy section: 10.1
Chapter 10
16.
Is the following a geometric series? Answer “Yes” or “No”.
2 + 1.2 + 0.72 + 0.432 + 0.2592 + …
Ans:
Yes
Learning Objectives: Determine whether or not a series is geometric.
difficulty: easy section: 10.1
17.
Is the following a geometric series? Answer “Yes” or “No”.
4 + 1 + 0.25 + 0.0625 + 0.015625 + …
Ans:
Yes
difficulty: easy section: 10.1
18.
Is the following a geometric series? Answer “Yes” or “No”.
3 + 6 + 9 + 12 + 15 + …
Ans:
Learning Objectives: Determine whether or not a series is geometric.
difficulty: easy section: 10.1
19.
Is the following a geometric series? Answer “Yes” or “No”.
5 7 9 11 13 …
17 17 17 17 17
+++++
Ans:
Learning Objectives: Determine whether or not a series is geometric.
difficulty: easy section: 10.1
20.
Which of the following are geometric series? Select all correct answers.
A)
23
2 2 2 2a a a+ + + +
B)
23
2 4 6 8a a a+ + + +
C)
2 2 3 3
2 2 2 2ak a k a k+ + + +
Ans: A, C Learning Objectives: Determine whether or not a series is geometric.
difficulty: easy section: 10.1
Chapter 10
21.
Find the sum
2
3
4
n
n
=
. Round to 2 decimal places.
Ans:
2.25
Learning Objectives: Compute sum of infinite geometric series.
difficulty: medium section: 10.1
22.
A yearly deposit of $10,000 is made into an account that pays 4.7% interest per year,
compounded annually. What is the balance right after the 8th deposit? Round to the
nearest cent.
Ans:
$94,472.51
market stabilization. difficulty: easy section: 10.2
23.
A couple wants to establish an annuity for retirement that will make annual payments of
$40,000 from an account that pays 5% interest per year, compounded annually. If the
payments are to start right now, how much should be deposited if they plan to live
indefinitely? Round to the nearest dollar.
Ans:
$840,000
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: medium section: 10.2
24.
A scholarship fund is set up to award 5 scholarships of $6000 each per year. The fund
earns 4% annual interest. How much money should be invested if the scholarships are
to be continued forever? Round to the nearest dollar.
Ans:
$780,000
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: medium section: 10.2
25.
The government gives a tax rebate totaling 4 billion dollars. The total additional
spending resulting from this tax rebate if everyone who receives the money spends 85%
of it is _____ billion dollars. Round to the nearest billion dollars.
Ans:
23
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: easy section: 10.2
26.
Find the market stabilization point if 5000 new units are manufactured each year and
15% of the total number of units in use fail each year.
Ans:
33,333
market stabilization. difficulty: easy section: 10.2
Chapter 10
27.
An employee accepts a job with a starting annual salary of $37,000 and a promised cost-
of-living increase of 2.5% per year. What are her total projected earnings over the next
7 years? Round to the nearest dollar.
Ans:
$279,255
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: easy section: 10.2
28.
An employee is offered two options: a fixed annual salary of $50,000, or $1 the first
month, $2 the second month, $4 the next month, and so on (doubling each month). If
the employee plans to work for two years, which option should he choose?
A)
The fixed annual salary.
B)
The salary that doubles each month.
Ans: B Learning Objectives: Use geometric series in financial models, including
annuities and market stabilization. difficulty: medium section: 10.2
29.
A farmer sells 10,000 pounds of potatoes per year. The current selling price is $0.20
per pound, but this price goes up 4% each year because of inflation. Predict the
farmer’s total earnings from potato sales over the next 10 years. Round to the nearest
dollar.
Ans:
$24,012
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: easy section: 10.2
30.
A school librarian estimates that 2% of the library’s books are either lost or damaged
each year and need to be pulled from the shelves. There are currently 250,000 books in
circulation, and the library adds 6,000 books each year. Is the library currently gaining
or losing books?
Ans:
gaining
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: medium section: 10.2
31.
A school librarian estimates that 2.5% of the library’s books are either lost or damaged
each year and need to be pulled from the shelves. There are currently 250,000 books in
circulation, and the library adds 6000 books each year. What will the stabilization point
be for the number of books in the library?
Ans:
240,000
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: easy section: 10.2
Chapter 10
32.
Twice a day, a patient takes a 25 mg tablet of a drug. At the end of a 12 hour period,
35% of the drug remains in the body. How many mg of the drug remain in the body
right after taking the 6th tablet? Round to 2 decimal places.
Ans:
38.39 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources. difficulty: easy
section: 10.3
33.
Twice a day, a patient takes a 25 milligrams tablet of a drug. At the end of a 12 hour
period, 25% of the drug remains in the body. How many milligrams of the drug remain
in the body at the steady state level, right after taking a tablet?
Ans:
33.33 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources. difficulty: easy
section: 10.3
34.
A new drug to control high blood pressure is found to have a half life of 2 days. A
patient takes two 50 milligram tablets of the drug at the same time each day. How
many milligrams of the drug are in the body after the 8th dose? Round to 2 decimal
places.
Ans:
320.08 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
section: 10.3
35.
A new drug to control high blood pressure is found to have a half life of 2 days. A
patient takes two 40 milligram tablets of the drug at the same time each day. How
many milligrams of the drug are in the body at the steady state right before taking the
tablets? Round to 2 decimal places.
Ans:
193.14 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources.
difficulty: medium section: 10.3
36.
A new drug to control high blood pressure is found to have a half life of 2 days. How
many milligrams of the drug would need to be given daily to achieve a steady state of
600 mg? Round to the nearest whole number.
Ans:
176 mg
Chapter 10
37.
Every evening, a person receives an 80 milligram injection of a drug. At the end of a
24 hour period, 40% of the drug remains in the body. How many milligrams of the
drug remain in the body right before the 5th injection? Round to the nearest whole
number.
Ans:
52 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources. difficulty: easy
section: 10.3
38.
Every evening, a person receives an 80 milligram injection of a drug. At the end of a
24 hour period, 45% of the drug remains in the body. How many milligrams of the
drug remain in the body right before receiving the injection at the steady state? Round
to the nearest whole number.
Ans:
65 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources.
difficulty: medium section: 10.3
39.
Every evening, a person receives an injection of a drug. At the end of a 24 hour period,
40% of the drug remains in the body. Complications arise if the level of the drug in the
body exceeds 140 milligrams. How many milligrams of the drug can be safely injected
each day? Round to the nearest whole number.
Ans:
84 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources. difficulty: hard
section: 10.3
40.
Each morning at breakfast, a person consumes 10 micrograms of a toxin found in a
pesticide, which leaves the body at a continuous rate of 5% per day. In the long run,
how many micrograms of the toxin have accumulated in the body right after breakfast
each day? Round to the nearest whole number.
Ans:
205 micrograms
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources.
difficulty: medium section: 10.3
Chapter 10
41.
At the end of the year 2000, the total reserves of a natural resource was approximately
500,000 m3. During 2001, 4000 m3 of the resource was consumed, and consumption
was predicted to increase 9% per year after that. Under these assumptions, how many
years (after the year 2000) will the resource last? Round to 1 decimal place.
Ans:
29.1 years
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources. difficulty: easy
section: 10.3
42.
At the end of the year 2000, the total reserves of a natural resource were approximately
500,000 m3. During 2001, 4000 m3 of the resource was consumed. A conservation
organization sets a goal to decrease usage by 2% per year. If they can accomplish this,
how many m3 of the resource will be left after 100 years? Round to the nearest whole
number.
Ans:
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources.
difficulty: medium section: 10.3
43.
A king estimates that there are 5000 tons of gold in his mines. Last year, 125 tons of
gold were mined and sold to finance the kingdom. How many years will it be before
the kingdom goes bankrupt if the amount of gold refined increases by 2% each year?
Round to 1 decimal place.
Ans:
29.7 years
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources. difficulty: easy
section: 10.3
44.
A person receives 35 milligrams of a drug each day, and the drug is metabolized and
eliminated at a continuous rate of 20% per day. Find the number of milligrams of the
drug in the person’s body in the long run using a geometric series, assuming the 35
milligrams is taken in a single oral dose each morning (find the quantity just after the
dose is taken). Round to 1 decimal place.
Ans:
193.1 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources.
difficulty: medium section: 10.3
Chapter 10
Page 10
45.
A person receives 55 milligrams of a drug each day, and the drug is metabolized and
eliminated at a continuous rate of 25% per day. Find the number of milligrams of the
drug in the person’s body in the long run using a differential equation, assuming the 55
milligrams is administered continuously throughout the day via a patch. Round to 1
decimal place.
46.
Find the sum of
2 10
15 15(1.1) 15(1.1) 15(1.1)+ + + +
.
Round to 2 decimal places. If it does not exist, enter “DNE”.
Ans:
277.97
Learning Objectives: Compute sum of finite geometric series. difficulty: easy
section: 10 review
47.
Find the sum of
2
33
25,000 25,000 25,000
44
+ + +
.
If it does not exist, enter “DNE”.
Ans:
100,000
Learning Objectives: Compute sum of finite geometric series. difficulty: easy
section: 10 review
48.
Find the sum of
0.6 1.2 2.4 4.8− + − +
.
If it does not exist, enter “DNE”.
Ans:
Learning Objectives: Compute sum of infinite geometric series. difficulty: easy
section: 10 review
49.
Use the fact that
0.367367367... 0.367 0.000367 0.000000367= + + +
and your
knowledge of geometric series to find a fraction equal to
0.367367367...
Ans:
367/999
Learning Objectives: Compute sum of infinite geometric series.
difficulty: medium section: 10 review
Ans:
220.0 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources.
difficulty: medium section: 10.3
Chapter 10
50.
A sweepstakes offered a grand prize of $500,000 per year for 5 years. Suppose all
payments are made into a savings account earning 4.5% interest a year, compounded
annually. How much money will be in the account after 5 years? Round to the nearest
dollar.
Ans:
$2,735,355
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: medium section: 10 review
51.
The government gives a tax rebate totalling 6 billion dollars. If everyone who receives
the rebate spends 90% of it, the total additional spending resulting from this tax rebate is
_____ billion dollars. Round to the nearest dollar.
Ans:
54
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: medium section: 10 review
52.
An account earns 5% interest per year, compounded annually. Suppose payments of
$6000 each are to be made once a year from the account for 10 years, starting now.
How much must be deposited now to cover these payments? Round to the nearest
dollar.
Ans:
$48,647
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: medium section: 10 review
53.
Find the market stabilization point if 300 new items are manufactured each year and
20% of the total number of items in use fail each year.
Ans:
1500
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: medium section: 10 review
54.
A person takes 350 milligrams of a pain killer every 4 hours. If the pain killer has a
half life of 3 hours, how many milligrams of the drug are in the body after 24 hours
(right after the 6th dose)? Round to 1 decimal place.
Ans:
578.0 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources. difficulty: easy
section: 10 review
Chapter 10
55.
A person takes 200 milligrams of a pain killer every 4 hours. If the pain killer has a
half life of 3 hours, how many milligrams of the drug are in the body in the long run,
right after each dose? Round to 1 decimal place.
Ans:
331.6 mg
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources.
difficulty: medium section: 10 review
56.
Every day a person consumes 6 micrograms of a toxin with his lunch. The toxin leaves
the body at a continuous rate of 8% per day. In the long run, how many micrograms of
the toxin are in the body right after lunch? Round to the nearest whole number.
Ans:
78 micrograms
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources.
difficulty: medium section: 10 review
57.
At the end of the year 2004, the total reserve of a mineral was 235,000 m3. In the year
2005, 4000 cubic meters of the mineral was consumed. How many cubic meters of the
mineral will remain in 2054 if consumption decreases by 4% each year? Round to the
nearest whole number.
Ans:
Learning Objectives: Use geometric series to model accumulation or depletion of
difficulty: medium section: 10 review
58.
A radioactive isotope is released into the air as an industrial by-product. This isotope is
not very stable due to radioactive decay. Two-thirds of the original radioactive material
loses its radioactivity after each month. If 11 grams of this isotope are released into the
atmosphere at the end of the first and every subsequent month, how many grams
radioactive material are in the atmosphere at the end of the twelfth month? Round to 1
decimal place.
Ans:
16.5 g
difficulty: medium section: 10 review
Chapter 10
59.
A radioactive isotope is released into the air as an industrial by-product. This isotope is
not very stable due to radioactive decay. Two-thirds of the original radioactive material
loses its radioactivity after each month. If 10 grams of this isotope are released into the
atmosphere at the end of the first and every subsequent month and the situation goes on
ad infinitum, how many grams radioactive material are in the atmosphere at the end of
each month in the long run?
Ans:
15 g
Learning Objectives: Use geometric series to model accumulation or depletion of
natural quantities, including drugs, toxins, and natural resources. difficulty: easy
section: 10 review
60.
Find the value of
Ans:
0.80
Learning Objectives: Compute sum of finite geometric series. difficulty: medium
section: 10 review
2 100
4 4 4
…
9 9 9
+ + +
to 2 decimal places.
61.
Find the value of the infinite product
1/8 1/16 1/32 1/64 …e e e e
to 3 decimal places.
Ans:
1.284
Learning Objectives: Compute sum of infinite geometric series.
difficulty: medium section: 10 review
62.
Suppose the government spends $3 million on highways. Some of this money is earned
by the highway workers who in turn spend $1,500,000 on food, travel, and
entertainment. This causes $750,000 to be spent by the workers in the food, travel, and
entertainment industries. This $750,000 causes another $375,000 to be spent; the
$375,000 causes another $187,500 to be spent, and so on. (Notice that each expenditure
is half the previous one.) Assuming that this process continues forever, how many
million dollars in total spending is generated by the original $3 million expenditure?
Ans:
6
Learning Objectives: Compute sum of infinite geometric series.
difficulty: medium section: 10 review
63.
Does the infinite series
1/3 2/3 4/3
7 7 7 7
76
6 6 6
+ + + + +
converge or diverge?
Ans:
converges
Learning Objectives: Compute sum of infinite geometric series. difficulty: easy
section: 10 review
Chapter 10
64.
Find the sum of the first 8 terms of the series
3/2 2
7 7 7 7
76
666
+ + + + +
. Round to 2
decimal places.
Ans:
11.82
Learning Objectives: Compute sum of infinite geometric series.
difficulty: medium section: 10 review
65.
A ball is dropped from a height of 16 feet and bounces. Each bounce is
4
7
of the height
of the bounce before. Find the total vertical feet the ball has traveled when it hits the
floor for the 4th time. Round to 2 decimal places.
Ans:
50.71 feet
Learning Objectives: Use geometric series in financial models, including annuities and
market stabilization. difficulty: medium section: 10 review
66.
Is
2 2 3 3
2 4 6 8ak a k a k+ + + +
a geometric series?
Ans:
no
Learning Objectives: Determine whether or not a series is geometric.
difficulty: easy section: 10 review