Geometric Series Full Test Bank Ch.10 7th Edition

Applied Calculus, 7e (Hughes-Hallett)

Chapter 10 Geometric Series

10.1 Geometric Series

1) Find the sum of

2 + + + ∙∙∙ + .

If it does not exist, enter "DNE".

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

2) Find the sum of

30 + 30(1.02) + 30 + 30 + ∙∙∙.

If it does not exist, enter "DNE".

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Compute sum of infinite geometric series.

3) Find the sum of

400 + 200 + 100 + 50 + ∙∙∙.

If it does not exist, enter "DNE".

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Compute sum of infinite geometric series.

4) Find the sum of

1 - + + + ∙∙∙.

If it does not exist, enter "DNE".

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Compute sum of infinite geometric series.

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 12th deposit? Round to the nearest dollar.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

6) Consider the sum S = 10 + 10(0.3) + 10 + 10 + ∙∙∙.

A. Calculate the partial sum . Round to 4 decimal places.

B. Calculate the infinite sum S. Round to 4 decimal places.

A. 14.2856

B. 14.2857

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.; Compute sum of infinite geometric series.

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 18th dose? Round to 2 decimal places.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

8) Does the infinite series 2 + + + + + ∙∙∙ converge or diverge?

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Compute sum of infinite geometric series.

9) Find for the series 2 + + + + + ∙∙∙. Round to 3 decimal places.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

10) Find the sum of the series . Round to 2 decimal places.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

11) If the sum of the series S = a + + + + is , what is a?

Diff: 3 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

12) A ball is dropped from a height of 15 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.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

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 has a bounce of 3/4 the height of the bounce before and is dropped from a height of 15 feet. Which ball bounces a greater total vertical distance?

A) the tennis ball B) the superball

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

14) Is the following a geometric series? Answer "Yes" or "No".

4 + 24 + 144 + 864 + 5184 + ∙∙∙

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Determine whether or not a series is geometric.

15) Is the following a geometric series? Answer "Yes" or "No".

+ + + + + ∙∙∙

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Determine whether or not a series is geometric.

16) Is the following a geometric series? Answer "Yes" or "No".

3 + 0.6 + 0.12 + 0.024 + 0.0048 + ∙∙∙

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Determine whether or not a series is geometric.

17) Is the following a geometric series? Answer "Yes" or "No".

4 + 2 + 1 + 0.5 + 0.25 + ∙∙∙

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Determine whether or not a series is geometric.

18) Is the following a geometric series? Answer "Yes" or "No".

3 + 8 + 13 + 18 + 23 + ∙∙∙

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Determine whether or not a series is geometric.

19) Is the following a geometric series? Answer "Yes" or "No".

+ + + + + ∙∙∙

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Determine whether or not a series is geometric.

20) Which of the following are geometric series? Select all correct answers.

A) 3 + 3a + 3 + 3 + ∙∙∙

B) 3 + 6a + 9 + 12 + ∙∙∙

C) 3 + 3ak + 3 + 3 + ∙∙∙

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Determine whether or not a series is geometric.

21) Find the sum . Round to 2 decimal places.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of infinite geometric series.

22) Find the value of + + ... + to 2 decimal places.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

23) Find the value of the infinite product ∙ ∙ ∙ ∙ ... to 3 decimal places.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of infinite geometric series.

24) Does the infinite series 4 + + + + + ... converge or diverge?

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Compute sum of infinite geometric series.

25) Find the sum of the first 6 terms of the series 4 + + + + + ∙∙∙. Round to 2 decimal places.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

26) A ball is dropped from a height of 14 feet and bounces. Each bounce is of the height of the bounce before. Find the total vertical feet the ball has traveled when it hits the floor for the time. Round to 2 decimal places.

Diff: 2 Var: 1

Section: 10.1

Learning Objectives: Compute sum of finite geometric series.

27) Is 6 + 12ak + 18 + 24 + ∙∙∙ a geometric series?

Diff: 1 Var: 1

Section: 10.1

Learning Objectives: Determine whether or not a series is geometric.

10.2 Applications to Business and Economics

1) 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 before the deposit? Round to the nearest cent.

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

2) A couple wants to establish an annuity for retirement that will make annual payments of $40,000 from an account that pays 10% interest per year, compounded annually. If the payments are to start right now, how much should be deposited if they plan to live 25 years? Round to the nearest dollar.

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

3) A scholarship fund is set up to award 5 scholarships of $5000 each per year. The fund earns 3% annual interest. How much money should be invested if the scholarships are to be continued forever? Round to the nearest dollar.

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

4) 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.

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

5) Find the market stabilization point if 5000 new units are manufactured each year and 10% of the total number of units in use fail each year.

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

6) An employee accepts a job with a starting annual salary of $30,000 and a promised cost-of-living increase of 2.5% per year. What are her total projected earnings over the next 5 years? Round to the nearest dollar.

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

7) 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 one year, which option should he choose?

A) The salary that doubles each month.

B) The fixed annual salary.

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

8) 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 15 years. Round to the nearest dollar.

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

9) A school librarian estimates that 3% 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?

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

10) 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?

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

11) Find the sum of

15 + 15(1.25) + 15 + ∙∙∙ + 15.

Round to 2 decimal places. If it does not exist, enter "DNE".

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Compute sum of finite geometric series.

12) Find the sum of

15,000 + 15,000 + 15,000 + ∙∙∙.

If it does not exist, enter "DNE".

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Compute sum of infinite geometric series.

13) Find the sum of

0.8 - 1.6 + 3.2 - 6.4 + ∙∙∙.

If it does not exist, enter "DNE".

Diff: 1 Var: 1

Section: 10.2

Learning Objectives: Compute sum of infinite geometric series.

14) Use the fact that 0.157157157... = 0.157 + 0.000157 + 0.000000157 + ∙∙∙ and your knowledge of geometric series to find a fraction equal to 0.157157157...

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Compute sum of infinite geometric series.

15) A sweepstakes offered a grand prize of $500,000 per year for 10 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 10 years? Round to the nearest dollar.

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

16) The government gives a tax rebate totaling 6 billion dollars. If everyone who receives the rebate spends 85% of it, the total additional spending resulting from this tax rebate is ________ billion dollars. Round to the nearest dollar.

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

17) An account earns 5% interest per year, compounded annually. Suppose payments of $4000 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.

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

18) 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.

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

19) Suppose the government spends $2.5 million on highways. Some of this money is earned by the highway workers who in turn spend $1,250,000 on food, travel, and entertainment. This causes $625,000 to be spent by the workers in the food, travel, and entertainment industries. This $625,000 causes another $312,500 to be spent; the $312,500 causes another $156,250 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 $2.5 million expenditure?

Diff: 2 Var: 1

Section: 10.2

Learning Objectives: Use geometric series in financial models, including annuities and market stabilization.

10.3 Applications to the Natural Sciences

1) 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 tablet? Round to 2 decimal places.

Diff: 1 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

2) Twice a day, a patient takes a 25 milligrams tablet of a drug. At the end of a 12 hour period, 35% 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?

Diff: 1 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

3) 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 dose? Round to 2 decimal places.

Diff: 1 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

4) 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 at the steady state right before taking the tablets? Round to 2 decimal places.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

5) 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 700 mg? Round to the nearest whole number.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

6) 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 injection? Round to the nearest whole number.

Diff: 1 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

7) 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 after receiving the injection at the steady state? Round to the nearest whole number.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

8) 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 150 milligrams. How many milligrams of the drug can be safely injected each day? Round to the nearest whole number.

Diff: 3 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

9) Each morning at breakfast, a person consumes 7 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.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

10) At the end of the year 2000, the total reserves of a natural resource was approximately 500,000 . During 2001, 3000 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.

Diff: 1 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

11) At the end of the year 2000, the total reserves of a natural resource were approximately 500,000 . During 2001, 3000 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 of the resource will be left after 100 years? Round to the nearest whole number.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

12) A king estimates that there are 5000 tons of gold in his mines. Last year, 200 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.

Diff: 1 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

13) A person receives 50 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 50 milligrams is taken in a single oral dose each morning (find the quantity just after the dose is taken). Round to 1 decimal place.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

14) A person receives 35 milligrams of a drug each day, and the drug is metabolized and eliminated at a continuous rate of 30% 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 35 milligrams is administered continuously throughout the day via a patch. Round to 1 decimal place.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

15) 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 dose)? Round to 1 decimal place.

Diff: 1 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

16) 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.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

17) Every day a person consumes 5 micrograms of a toxin with his lunch. The toxin leaves the body at a continuous rate of 9% per day. In the long run, how many micrograms of the toxin are in the body right before lunch? Round to the nearest whole number.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

18) At the end of the year 2004, the total reserve of a mineral was 235,000 . In the year 2005, 3500 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.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

19) 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 8 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.

Diff: 2 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

20) 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?

Diff: 1 Var: 1

Section: 10.3

Learning Objectives: Use geometric series to model accumulation or depletion of natural quantities, including drugs, toxins, and natural resources.

© (2022) John Wiley & Sons, Inc. All rights reserved. Instructors who are authorized users of this course are permitted to download these materials and use them in connection with the course. Except as permitted herein or by law, no part of these materials should be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise.