Deck 10: Geometric Series

Question 1

Find the sum of .
If it does not exist, enter "DNE".

Answer 1

1200

Question 2

Does the infinite series converge or diverge?

Answer 2

converges

Question 3

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 15^th dose? Round to 2 decimal places.

Answer 3

53.33

Question 4

Is the following a geometric series? Answer "Yes" or "No".
2 + 1.2 + 0.72 + 0.432 + 0.2592 + ...

Question 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 11^th deposit? Round to the nearest dollar.

Question 6

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

Question 7

Find the sum of .
If it does not exist, enter "DNE".

Question 8

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

Question 9

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 n^th time, and use it to find the total vertical distance the ball has traveled when it hits the floor for the 4^th time. Round to 2 decimal places.

Question 10

Is the following a geometric series? Answer "Yes" or "No".
3 + 6 + 9 + 12 + 15 + ...

Question 11

Is the following a geometric series? Answer "Yes" or "No".
4 + 1 + 0.25 + 0.0625 + 0.015625 + ...

Question 12

Find the sum . Round to 2 decimal places.

Question 13

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

Question 14

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

A)
B)
C)

Question 15

Find for the series Round to 3 decimal places.

Question 16

Find the sum of .
If it does not exist, enter "DNE".

Question 17

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

Question 18

Is the following a geometric series? Answer "Yes" or "No".
2 + 4 + 8 + 16 + 32 + ...

Question 19

Find the sum of .
If it does not exist, enter "DNE".

Question 20

If the sum of the series is , what is a?

Question 21

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?

Question 22

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 6^th tablet? Round to 2 decimal places.

Question 23

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.

Question 24

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.

Question 25

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 8^th dose? Round to 2 decimal places.

Question 26

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 8^th deposit? Round to the nearest cent.

Question 27

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.

Question 28

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?

Question 29

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.

Question 30

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.

Question 31

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?

Question 32

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.

Question 33

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.

Question 34

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.

Question 35

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 5^th injection? Round to the nearest whole number.

Question 36

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.

Question 37

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.

Question 38

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.

Question 39

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.

Question 40

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.

Question 41

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.

Question 42

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.

Question 43

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.

Question 44

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.

Question 45

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.

Question 46

Find the sum of .
Round to 2 decimal places. If it does not exist, enter "DNE".

Question 47

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.

Question 48

Find the value of to 2 decimal places.

Question 49

Find the value of the infinite product to 3 decimal places.

Question 50

Find the sum of .
If it does not exist, enter "DNE".

Question 51

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?

Question 52

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.

Question 53

Find the sum of .
If it does not exist, enter "DNE".

Question 54

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.

Question 55

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.

Question 56

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.

Question 57

Use the fact that and your knowledge of geometric series to find a fraction equal to

Question 58

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.

Question 59

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.

Question 60

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 6^th dose)? Round to 1 decimal place.

Question 61

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?

Question 62

Does the infinite series converge or diverge?

Question 63

A ball is dropped from a height of 16 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 4^th time. Round to 2 decimal places.

Question 64

Is a geometric series?

Question 65

Find the sum of the first 8 terms of the series . Round to 2 decimal places.