Chapter Ten
GEOMETRIC SERIES
Contents
Repeated Deposits into a Savings Account
Sum of a Finite Geometric Series
Sum of an Infinite Geometric Series
10.2 Applications to Business and Economics
10.3 Applications to the Natural Sciences
Accumulation of Toxins in the Body
Depletion of Natural Resources
10.1 GEOMETRIC SERIES
Repeated Drug Dosage
Malaria is a parasitic infection transmitted by mosquito bites, mainly in tropical areas of the world. The disease has existed since ancient times, and currently there are hundreds of millions of cases each year, with millions of deaths. In the 1640s, the Jesuits in Peru introduced the bark of the cinchona tree to the West as the first treatment for malaria. The drug quinine is the active ingredient in the bark, and it is still used today.
Suppose a person is given a 50-mg dose of quinine at the same time every day for the treatment of malaria. After the first dose, the person has 50 mg of quinine in the body. What about after the second dose? Each day, the person's body metabolizes some of the quinine so that, after one day, 23% of the original amount remains. After the second dose, the amount of quinine in the body is the amount from the second dose (50 mg) plus the remnants of the first dose (that is, 50·0.23 = 11.5 mg) for a total of 61.5 mg.
Let Qn represent the quantity, in mg, of quinine in the body right after the nth dose. Then
Notice that we can multiply out the expression for Q3 to show the contributions of the first and second dose separately:
so we have
The multiplied-out form of Q3 enables us to guess formulas for later values of Qn:
The values of Q6 and Q10 suggest that the quantity is stabilizing at around 64.9 mg. See Figure 10.1.
Figure 10.1: Quantity of quinine levels off
Figure 10.2: Bank balance grows without bound
Repeated Deposits into a Savings Account
People who save money often do so by putting a fixed amount aside regularly. Suppose $1000 is deposited every year in a savings account earning 5% interest a year, compounded annually. Let Bn represent the balance, in dollars, in the account right after the nth deposit. Then
As before, we multiply out the expression for B3 to show the contributions of the first and second deposits separately:
The multiplied-out formula for B3 enables us to guess formulas for B6 and B10. Evaluating gives
Notice that the balance is growing without bound. See Figure 10.2.
Finite Geometric Series
In the two previous examples, we encountered sums of the form a+ar+ar2+ar3+···+ar8+ar9. Such a sum is called a finite geometric series. A geometric series is a sum in which each term is a constant multiple of the preceding one. The first term is a, and the constant multiplier, or common ratio, of successive terms is r.
A finite geometric series with n terms has (for n a positive integer) the form
Sum of a Finite Geometric Series
In the quinine example, suppose we want to find Q40, the quantity of quinine in the body after 40 doses:
To calculate Q40, we could add these 40 terms. Fortunately, there is a better way.
We write Sn for the sum of the first n terms of the series, that is, up to the term arn−1:
Multiplying both sides by r gives
Now subtract rSn from Sn, so all the terms except two on the right cancel, giving
so
Provided r ≠ 1, we can solve for Sn. The result is called a closed form for Sn.
Note that the value of n in the formula is the number of terms in the sum Sn.
| Example 1 | In the quinine example, calculate and interpret Q40 and Q100. |
| Solution | We saw earlier that
This is a finite geometric series with a = 50 and r = 0.23. Using the formula for the sum with n = 40, we have
The amount of quinine in the body right after the 40th dose is 64.935 mg. Similarly, using n = 100, we have
Right after the 100th dose, the amount of quinine in the body is still 64.935 mg. To three decimal places, the amount appears to have stabilized. |
| Example 2 | In the bank deposit example, calculate and interpret B40 and B100. |
| Solution | We have
This is a finite geometric series with a = 1000 and r = 1.05. The formula for the sum with n = 40 gives
The balance in the account right after the 40th deposit is $120,799.77. Similarly, using n = 100, we have
Right after the 100th deposit, the balance in the account is $2,610,025.16. Compound interest has increased the $100,000 investment to over $2 million. |
Infinite Geometric Series
Suppose a finite geometric series has n terms in it. What happens as n → ∞? We get an infinite geometric series that goes on forever.
An infinite geometric series has the form
The “. . .” at the end of the series tells us that the series goes on forever—it has an infinite number of terms.
Sum of an Infinite Geometric Series
Given an infinite geometric series
we call the sum of the first n terms a partial sum, written Sn. To calculate Sn, we use the formula
What happens to Sn as n → ∞? It depends on the value of r. If |r| < 1, that is, −1 < r < 1, then rn → 0 as n → ∞, so as n → ∞,
Thus, provided |r| < 1, as n → ∞ the partial sums Sn approach a limit of a/(1 − r). When this happens, we define the sum of the infinite geometric series to be that limit and say the series converges to a/(1 − r).
For |r| < 1, the sum of the infinite geometric series is given by
If, on the other hand, |r| > 1, then rn and the partial sums have no limit as n → ∞ (if a ≠ 0). In this case, we say the series diverges. If r > 1, the terms in the series become larger and larger in magnitude, and the partial sums diverge to +∞ if a > 0, or to −∞ if a < 0. If r < −1, the terms become larger in magnitude, the partial sums oscillate as n → ∞, and the series diverges.
What happens if r = 1? The series is
so if a ≠ 0, the partial sums grow without bound, and the series does not converge. If r = −1, the series is
and, if a ≠ 0, the partial sums oscillate between a and 0, and the series does not converge.
| Example 3 | For each of the following infinite series, find the first three partial sums and the sum (if it exists):
(a) 10 + 10(0.75) + 10(0.75)2 + · · · (b) 250 + 250(1.2) + 250(1.2)2 + · · · |
| Solution | (a) This is an infinite geometric series with a = 10 and r = 0.75. The first three partial sums are:
Since |r| < 1, the series converges and the sum is
If we find partial sums for larger and larger n, they get closer and closer to 40. (See Problem 23.) (b) This is an infinite geometric series with a = 250 and r = 1.2. The first three partial sums are:
Since r > 1, the series diverges, and the partial sums grow without bound. (See Problem 22.) |
| Example 4 | Suppose 50-mg doses of quinine are taken daily forever. Find the long-run quantities of quinine in the body right after and right before a dose is given. |
| Solution | Since quinine is given forever, from Example 1 we know that the long-run quantity of quinine, right after a dose, is given by
This is an infinite geometric series with a = 50 and r = 0.23. Since −1 < r < 1, the series converges to a finite sum given by
The long-run quantity of quinine in the body right after a dose is 64.935 mg. The quantity levels off to this value in Figure 10.1. What is the long-run quantity of quinine right before a dose? Since a dose is 50 mg, the quantity of quinine in the body right before a dose is 64.935 − 50 = 14.935 mg. Thus, in the long run, the quinine level oscillates between 15 mg and 65 mg. |
| Example 5 | Suppose that $1000 a year is deposited forever into the bank account in Example 2. Does the balance in the account stabilize at a fixed amount? Explain. |
| Solution | Since the deposits are made forever, the account balance right after a deposit is represented by
This is an infinite geometric series with a = 1000 and r = 1.05. Since r is larger than 1, the series diverges. This makes sense, since if you keep depositing $1000 in an account, your balance grows without bound, even if you don't earn interest. This matches Figure 10.2 and Example 2. |
Sometimes the terms of the geometric series may need to be rewritten before the values of a and r can be identified.
| Example 6 | Find the sum of the geometric series.
|
| Solution | (a) The infinite series 1 + 0.1 + 0.01 + 0.001 + · · · can be rewritten as
(b) The finite series
With a = 4 and r = 1/2 we have
|
can be rewritten as
Problems for Section 10.1
1. Find the sum of the following series in two ways: by adding terms and by using the geometric series formula.
2. Find the sum of the following series in two ways: by adding terms and by using the geometric series formula.
In Problems 3–9, decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not.
3. 
4. 
5. 
6. 
7. 1 + 2z + (2z)2 + (2z)3 + · · ·
8. 1 + x + 2x2 + 3x3 + 4x4 + · · ·
9. y2 + y3 + y4 + y5 + · · ·
In Problems 10–21, find the sum, if it exists.
10. 5 + 5 · 3 + 5 · 32 + · · · + 5 · 312
11. 100 + 100(0.85) + 100(0.85)2 + · · · + 100(0.85)10
12. 1000 + 1000(1.05) + 1000(1.05)2 + · · ·
13. 75 + 75(0.22) + 75(0.22)2 + · · ·
14. 20 + 20(1.45) + 20(1.45)2 + · · · + 20(1.45)14
15. 500(0.4) + 500(0.4)2 + 500(0.4)3 + · · ·
16. 31500 + 6300 + 1260 + 252 + · · ·
17. 
18. 1000 + 1500 + 2250 + 3375 + 5062.5 + · · ·
19. 200 + 100 + 50 + 25 + 12.5 + · · ·
20. 
21. 
22. In Example 3(b) on page 467, we found partial sums for the geometric series with a = 250 and r = 1.2. Find the partial sums Sn for n = 5, 10, 15, 20. As n gets larger, do the partial sums appear to grow without bound, as expected if r > 1?
23. In Example 3(a), we found partial sums of the geometric series with a = 10 and r = 0.75 and showed that the sum of this series is 40. Find the partial sums Sn for n = 5, 10, 15, 20. As n gets larger, do the partial sums appear to be approaching 40?
24. A repeating decimal can always be expressed as a fraction. This problem shows how writing a repeating decimal as a geometric series enables you to find the fraction.
(a) Write the repeating decimal 0.232323. . . as a geometric series using the fact that 0.232323 . . . = 0.23 + 0.0023 + 0.000023 + · · ·.
(b) Use the formula for the sum of a geometric series to show that 0.232323 . . . = 23/99.
25. Every month, $500 is deposited into an account earning 0.1% interest a month, compounded monthly.
(a) How much is in the account right after the 6th deposit? Right before the 6th deposit?
(b) How much is in the account right after the 12th deposit? Right before the 12th deposit?
26. Each year, a family deposits $5000 into an account paying 1.25% interest per year, compounded annually. How much is in the account right after the 25th deposit?
27. A smoker inhales 0.4 mg of nicotine from a cigarette. After one hour, 71% of the nicotine remains in the body. If a person smokes one cigarette every hour beginning at 7 am, how much nicotine is in the body right after the 11 pm cigarette?
28. Each morning, a patient receives a 25 mg injection of an anti-inflammatory drug, and 40% of the drug remains in the body after 24 hours. Find the quantity in the body:
(a) Right after the 3rd injection.
(b) Right after the 6th injection.
(c) In the long run, right after an injection.
29. In Example 4 on page 468, we saw that if 50 mg of quinine is given every 24 hours, the long-run quantity of quinine in the body is about 65 mg right after a dose and about 15 mg right before a dose. The concentration of quinine in the body is measured in milligrams of quinine per kilogram of body weight. To be effective, the average concentration of quinine in the body must be at least 0.4 mg/kg. Concentrations above 3.0 mg/kg are not safe.
(a) Estimate the average quantity of quinine in the body in the long run by averaging the long-run quantities of quinine in the body right after a dose and right before a dose.
(b) Find the average concentration for a person weighing 70 kilograms. Is this treatment safe and effective for such a person?
(c) For what range of weights would this treatment produce a long-run average concentration that is
(i) Too low?
(ii) Unsafe?
30. A ball is dropped from a height of 10 feet and bounces. Each bounce is 
of the height of the bounce before. Thus, after the ball hits the floor for the first time, the ball rises to a height of 10() = 7.5 feet, and after it hits the floor for the second time, it rises to a height of 7.5() = 10()2 = 5.625 feet.
(a) Find an expression for the height to which the ball rises after it hits the floor for the nth time.
(b) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, third, and fourth times.
(c) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the nth time. Express your answer in closed form.
10.2 APPLICATIONS TO BUSINESS AND ECONOMICS
Annuities
An annuity is a sequence of equal payments made at regular intervals. We can use the sum of a geometric series to calculate the total value of an annuity.
| Example 1 | An annuity pays $5000 every year into an account that earns 7% interest per year, compounded annually. What is the balance in the account right after the 10th deposit? |
| Solution | The 10th deposit contributes $5000 to the balance. The previous deposit has earned interest for a year, so it contributes $5000(1.07). The deposit the year before that has earned interest for two years, so it contributes $5000(1.07)2. Continuing, we see that
This sum is a finite geometric series with a = 5000 and r = 1.07. We use the formula for the sum with n = 10:
The balance in the account right after the 10th deposit is $69,082.24. |
Present Value of an Annuity
The present value of an annuity is the amount of money that must be deposited today to make a series of fixed payments in the future. How can we compute this present value? We begin by considering a single payment. Suppose a payment of $1000 is to be made three years in the future from an account paying interest at a rate of 8% per year, compounded annually. The present value is the amount P such that
so we have
To find the present value of four payments of $1000 at the same 8% interest, one made now, one in one year's time, one in two years, and one in three years, we add their present values:
This pattern allows us to find the present value of any annuity, as in the following example.
| Example 3 | The annuity in Example 2 now makes annual payments of $10,000 in perpetuity (that is, forever), rather than just twenty times. What is the present value of this annuity? |
| Solution | Since the payments are made forever, the present value is given by the infinite sum:
This is an infinite geometric series with a = 10,000 and r = (1.08)−1 = 0.925926. Since −1 < r < 1, this series converges to a finite sum. We have
The present value of this annuity in perpetuity is $135,000. Notice that the amount needed to make annual payments forever is only about $29,000 more than the amount needed to make 20 annual payments. This is the power of compound interest. |
The Multiplier Effect
A government decides to give a tax rebate to stimulate the economy. What is the total effect of the rebate on spending? Each individual who receives a tax rebate spends some proportion of the additional income, and this money becomes additional income for a second individual, who spends a proportion of it, providing additional income for a third individual, and so on. The total effect of the rebate on the economy is much larger than the size of the rebate itself; this is called the multiplier effect.
| Example 4 | A government gives tax rebates totaling 3 billion dollars. Everyone who receives money spends 75% of it and saves the other 25%. Find the total additional spending resulting from this tax rebate. |
| Solution | The additional spending refers to all the additional money spent by consumers as a result of this tax rebate. The recipients of the 3 billion dollars spend 75% of what they receive, that is 3(0.75) = 2.25 billion dollars. The recipients of this 2.25 billion dollars spend 75% of that, or 2.25(0.75) = 1.6875 billion dollars. The recipients of this money spend 75% of this amount, and so on. Thus,
Total additional spending = 2.25 + 2.25(0.75) + 2.25(0.75)2 + 2.25(0.75)3 + · · · billion dollars. (Notice that the initial amount spent was 2.25, not 3 billion dollars.) This is an infinite geometric series with a = 2.25 and r = 0.75. Since −1 < r < 1, this series converges to a finite sum:
Thus, a 3 billion dollar tax rebate generates 9 billion dollars of additional spending. |
Market Stabilization
Each year, a manufacturer produces a fixed number of units of a product and each year a fixed percentage of these units (regardless of age) fail or go out of use. The total number of units in use in the long run is called the market stabilization point.
| Example 5 | The US Mint produces about 7.4 billion pennies a year and about 5% of them are removed from circulation each year.1 Approximately how many pennies are in circulation? |
| Solution | We can estimate the number of pennies in circulation using geometric series if we assume (for example) that the yearly production takes place at the start of the year and that the 5% are removed at the end of the year. In any year, 7.4 billion pennies are produced, and 7.4(0.95) billion pennies remain from the previous year's production (since 5% went out of circulation). There are 7.4(0.95)2billion pennies remaining from those produced two years before, and so on. If N is the number of pennies in circulation, in billions, then
This is an infinite geometric series with a = 7.4 and r = 0.95. Since −1 < r < 1, the series converges and its sum is given by
Thus, the market has stabilized with about 148 billion pennies in circulation today. Of the 148 billion pennies in circulation, 5% or (148)(0.05) = 7.4 billion are removed each year, which exactly equals the number produced each year. |
Problems for Section 10.2
1. A yearly deposit of $1000 is made into a bank account that pays 2% interest per year, compounded annually. What is the balance in the account right after the 20th deposit? How much of the balance comes from the annual deposits and how much comes from interest?
2. Annual deposits of $2000 are made into an account paying 2.5% interest per year, compounded continuously. What is the balance in the account right after and right before the 5th deposit?
3. An annuity makes annual payments of $50,000, starting now, from an account paying 3.65% interest per year, compounded annually. Find the present value of the annuity if it makes
(a) Ten payments
(b) Payments in perpetuity
4. Twenty annual payments of $5000 each, with the first payment one year from now, are to be made from an account earning 2% per year, compounded annually. How much must be deposited now to cover the payments?
5. What is the present value of an annuity that pays $20,000 each year, forever, starting today, from an account that pays 1% interest per year, compounded annually?
6. An annuity earning 0.25% per month, compounded monthly, is to make 36 monthly payments of $1000 each, starting now. What is the present value of this annuity?
7. A donor sets up an endowment to fund an annual scholarship of $10,000. The endowment earns 3% interest per year, compounded annually. Find the amount that must be deposited now if the endowment is to fund one award each year, with one award now and continuing
(a) Until twenty awards have been made
(b) Forever
8. An employee accepts a job with a starting salary of $30,000 and a cost-of-living increase of 2% every year for the next 10 years. What is the employee's salary at the start of the 11th year? What are her total earnings during the first 10 years?
9. An employer pays you 1 penny the first day you work and doubles your wages each day after that. Find your total earnings after working 7 days a week for
(a) One week
(b) Two weeks
(c) Three weeks
(d) Four weeks
10. Find the market stabilization point for a product if 10,000 new units of the product are manufactured at the start of each year and 25% of the total number of units in use fail at the end of each year.
11. Every year, a company sells 1000 units of a product while 20% of the total number in use fail. Assume sales are at the start of the year and failures are at the end of the year.
(a) Find the market stabilization point for this product.
(b) If the stabilization point is approached very slowly, the number of units in use may not get close to this value because market conditions change first. Make a table for Sn, the number of units in use right after the nth annual sale, for n = 5, 10, 15, 20, to see how rapidly this market approaches the stabilization point.
12. Since 2007, the Bureau of Engraving and Printing has produced about 11 million new $1 bills a day; worn bills are removed by Federal Reserve banks. There are about 9.2 billion dollar bills currently in circulation. Assuming that a fixed percentage of $1 bills are removed from circulation each day, use a geometric series to estimate this percentage.2
Problems 13–15 are about bonds, which are issued by a government to raise money. An individual who buys a $1000 bond gives the government $1000 and in return receives a fixed sum of money, called the coupon, every six months or every year for the life of the bond. At the time of the last coupon, the individual also gets back the $1000, or principal.
13. What is the present value of a $1000 bond which pays $50 a year for 10 years, starting one year from now? Assume the interest rate is 4% per year, compounded annually.
14. What is the present value of a $1000 bond which pays $50 a year for 10 years, starting one year from now? Assume the interest rate is 6% per year, compounded annually.
15. (a) What is the present value of a $1000 bond which pays $50 a year for 10 years, starting one year from now? Assume the interest rate is 5% per year, compounded annually.
(b) Since $50 is 5% of $1000, this bond is called a 5% bond. What does your answer to part (a) tell you about the relationship between the principal and the present value of this bond if the interest rate is 5%?
(c) If the interest rate is more than 5% per year, compounded annually, which is larger: the principal or the present value of the bond? Why is the bond then described as trading at a discount?
(d) If the interest rate is less than 5% per year, compounded annually, why is the bond described as trading at a premium?
16. To stimulate the economy in 2002, the government gave a tax rebate totaling 40 billion dollars. Find the total additional spending resulting from this tax rebate if everyone who receives money spends
(a) 80% of it
(b) 90% of it
17. To stimulate the economy in 2008, the government gave a tax rebate totaling 100 billion dollars. Find the total additional spending resulting from this tax rebate if everyone who receives money spends
(a) 80% of it
(b) 90% of it
18. In April 2009, economists predicted that each dollar in tax cuts would generate $3 in economic growth. What spending rate was assumed in arriving at this estimated multiplier effect?
19. This problem illustrates how banks create credit and can thereby lend out more money than has been deposited. Suppose that initially $100 is deposited in a bank. Experience has shown bankers that on average only 8% of the money deposited is withdrawn by the owner at any time. Consequently, bankers feel free to lend out 92% of their deposits. Thus $92 of the original $100 is loaned out to other customers (to start a business, for example). This $92 becomes someone else's income and, sooner or later, is redeposited in the bank. Thus 92% of $92, or $92(0.92) = $84.64, is loaned out again and eventually redeposited. Of the $84.64, the bank again loans out 92%, and so on.
(a) Find the total amount of money deposited in the bank as a result of these transactions.
(b) The total amount of money deposited divided by the original deposit is called the credit multiplier. Calculate the credit multiplier for this example and explain what this number tells us.
20. A person who deposits money in a bank account starts a long process described by the reserve-deposit ratio, r. For every dollar deposited, the bank keeps r dollars and lends 1 − r dollars to someone else, who deposits the loan in a bank account. The same fraction of the second deposit is loaned out, to be deposited in turn, and so on. If the initial deposit is N dollars, find the total value of the bank accounts generated by this deposit:
(a) After the second deposit
(b) After the third deposit
(c) If the process continues forever
10.3 APPLICATIONS TO THE NATURAL SCIENCES
Steady-State Drug Levels
A patient is given a drug at regular intervals. In Section 10.1, we saw that in the long run, the quantity of drug in the body varies between a maximum level right after a dose and a minimum level right before a dose. At this steady state, the quantity eliminated between doses is equal to one dose. (See Problem 19.)
| Example 1 | A child with an ear infection is given a 200-mg ampicillin tablet once every 4 hours. About 12% of the drug in the body at the start of a four-hour period is still there at the end of that period. What is the quantity of ampicillin in the body
(a) Right after taking the 3rd tablet? (b) Right after taking the 6th tablet? (c) At the steady state, right after and right before taking a tablet? |
| Solution | Let Qn be the quantity of ampicillin, in mg, in the body right after taking the nth tablet. Then
(a) Using the formula for the sum of a finite geometric series with a = 200, r = 0.12, and n = 3, we have
(b) Using n = 6, we have
(c) At the steady state, the quantity Q in mg, right after a tablet is taken is the sum of the infinite geometric series
Since r = 0.12 and −1 < r < 1, the series converges to a finite sum. At the steady state
The quantity of ampicillin in the body right before a dose is exactly 200 mg less than the quantity right after the dose, so at the steady state
Problem 4 shows how this relationship can be used to calculate Q, the long-run quantity of ampicillin right after a dose. |
Accumulation of Toxins in the Body
Toxins found in pesticides can get into the food chain and accumulate in people's bodies through the food they eat. We use a geometric series to calculate the total accumulation of toxin in the body.
| Example 3 | Every day, a person consumes 5 micrograms (μg) of a toxin, which leaves the body at a continuous rate of 2% per day. In the long run, how much toxin is in the body at the end of the day? |
| Solution | Since the toxin leaves the body at a continuous rate of 2% every day, the 5 μg amount consumed one day earlier has decayed to 5e−0.02, the 5 μg consumed two days earlier has decayed to 5e−0.02(2) = 5(e−0.02)2, and so on. We assume that the 5 μg consumed during the day has not yet decayed. In the long run, at the end of each day we have, in micrograms,
This is an infinite geometric series with a = 5 and r = e−0.02 = 0.9802. Since −1 < r < 1, the series converges and its sum is
In the long run, there are 252.5 micrograms of toxin in the body at the end of the day. |
Depletion of Natural Resources
Geometric series can be used to estimate how long a natural resource (such as oil) will last, assuming that usage levels increase at a constant percentage rate.
| Example 4 | At the end of 2008, world oil reserves were about 1950 billion barrels.3 During 2008, about 29.3 billion barrels of oil were consumed.4 Over the past decade, oil consumption has been increasing at about 1% per year.5 Assuming yearly oil consumption increases at this rate in the future, how long will the reserves last? |
| Solution | Under these assumptions, the oil used in 2009, in billions of barrels, is predicted to be 29.3(1.01). In 2010, we predict 29.3(1.01)2 billion barrels to be used, and 29.3(1.01)3 the next year, and so on. Thus, starting with 2009, the total quantity of oil used in n years, Qn, is
This is a finite geometric series with n terms where a = 29.3(1.01) and r = 1.01. (Notice that the last term can be written 29.3(1.01)(1.01)n−1.) The sum is
We want to find the value of n for which Qn reaches the total reserves, 1950. Solving gives
Taking logarithms and using ln(Ap) = p ln A, we have
Thus, if present consumption patterns are maintained, the world's oil supply will be exhausted in about 51 years. However, if consumption patterns change, the length of time until the reserves run out can be very different. Problem 3 concerns predictions using a 2.5% yearly increase and a 5.5% yearly decrease, the maximum and minimum figures for the past decade. |
Geometric Series and Differential Equations
In this section, we used geometric series to model drug levels; in Chapter 9, we used differential equations to model drug levels. How do we know whether to use a geometric series or a differential equation? The answer depends on whether the drug is given in discrete doses (such as a tablet each morning) or continuously (such as intravenously).
| Example 5 | A patient receives 25 mg of a drug each day, and the drug is metabolized and eliminated at a continuous rate of 10% per day. Find the quantity of drug in the patient's body in the long run:
(a) Using a geometric series, assuming the 25-mg dose of the drug is administered in a single injection each morning. (Find the quantity both before and after an injection is given.) (b) Using a differential equation, assuming the 25-mg dose of the drug is administered continuously throughout the day, using an intravenous drip. |
| Solution | (a) A 25-mg injection is given each day. Since the drug is metabolized at a continuous rate of 10% per day, the quantity remaining a day later is 25e−0.1 mg. The quantity remaining two days later is 25(e−0.1)2 mg. In the long run,
We use the formula for the sum of an infinite geometric series with a = 25 and r = e−0.1:
Since each injection is 25 mg, in the long run
(b) The drug is entering the body at the continuous rate of 25 mg per day and leaving at the continuous rate of 0.1 times the current level in the body. Thus, if Q is the quantity of drug in the body after t days, then
In Chapter 9, we saw that Q tends toward the equilibrium solution which occurs when dQ/dt = 0. Then
When the drug is given continuously, the quantity of the drug in the body levels off at 250 mg. |
Problems for Section 10.3
1. Every morning, a patient receives a 50-mg injection of a drug. At the end of a 24-hour period, 60% of the drug remains in the body. What quantity of drug is in the body
(a) Right after the 3rd injection?
(b) Right after the 7th injection?
(c) Right after an injection, at the steady state?
2. In 2010, world oil consumption was 87.4 million barrels per day,6 an increase of 3.1% from 2009. Assuming that consumption continues to decrease at the same percentage rate, make a table showing yearly consumption between 2010 and 2017, inclusive. Find the total quantity of oil consumed during this decade.
3. As in Example 4, assume that oil reserves at the end of 2008 were 1950 billion barrels and that consumption in 2008 was 29.3 billion barrels. What happens in the long run if oil consumption
(a) Decreases by 5.5% per year
(b) Increases by 2.5% per year
4. This problem gives another way of finding, Q, the long-run ampicillin quantity right after a dose in Example 1. (See page 474.) In the four hours after a dose, 200 mg of ampicillin must be excreted, as the next 200-mg tablet elevates the ampicillin quantity back to Q mg. Use this information to solve for Q.
5. A dose of 120 mg is taken by a patient at the same time every day. In one day, 30% of the drug is excreted.
(a) At the steady state, find the quantity of drug in the body right after a dose.
(b) Check that at the steady state, the quantity excreted in one day is equal to the dose.
6. A person with chronic pain takes a 30 mg tablet of morphine every 4 hours. The half-life of morphine is 2 hours.
(a) How much morphine is in the body right after and right before taking the 6th tablet?
(b) At the steady state, find the quantity of morphine in the body right after and right before taking a tablet.
7. At the same time every day, a patient takes 50 mg of the antidepressant fluoxetine, whose half-life is 3 days.
(a) What fraction of the dose remains in the body after a 24-hour period?
(b) What is the quantity of fluoxetine in the body right after taking the 7th dose?
(c) In the long run, what is the quantity of fluoxetine in the body right after a dose?
8. (a) An allergy drug with a half-life of 18 weeks is given in 100-mg doses once a week. At the steady state, find the quantity of the drug in the body right after a dose.
(b) The drug does not become effective until the quantity in the body right after a dose reaches 2000 mg. How many weeks after the first dose does the drug become effective?
9. Each day at lunch a person consumes 8 micrograms of a toxin found in a pesticide; the toxin is metabolized at a continuous rate of 0.5% per day. In the long run, how much of this toxin accumulates in the person's body? Give the quantities right after and right before lunch.
10. A cigarette puts 1.2 mg of nicotine into the body. Nicotine leaves the body at a continuous rate of 34.65% per hour, but more than 60 mg can be lethal. If a person smokes a cigarette with each of the following frequencies, find the long-run quantity of nicotine in the body right after a cigarette. Does the nicotine reach the lethal level?
(a) Every hour
(b) Every half hour
(c) Every 15 minutes
(d) Every 6 minutes
(e) Every 3 minutes
11. At the end of 2008, the total reserve of a mineral was 350,000 m3. In the year 2009, about 5000 m3 was used. Each year, consumption of the mineral is expected to increase by 8%. Under these assumptions, in how many years will all reserves of the mineral be depleted?
12. At the end of 2007, natural gas reserves were 180 trillion m3; during 2007, about 3 trillion m3 of natural gas were consumed.7 Estimate how long natural gas reserves will last if consumption increases at 2% per year.
13. Over the past decade, natural gas consumption has been increasing at between 0% and 5% a year. Using the data from Problem 12, estimate how long the natural gas reserves will last assuming the rate of increase is
(a) 0%
(b) 5%
14. We use 1500 kg of a mineral this year and consumption of the mineral is increasing annually by 4%. The total reserves of the mineral are estimated to be 120,000 kg. Approximately when will the reserves run out?
Problems 15–17 concern how long reserves of the mineral in Problem 11 last if usage patterns change. For example, as reserves get lower, substitutes may be developed.
15. How long will the reserves last if the annual increase in usage is 4%?
16. How long will the reserves last if the annual usage stays constant at 5000 m3 per year?
17. How long will the reserves last if the usage decreases each year by 4%?
18. (a) A dose D of a drug is administered at intervals equal to the half-life. (That is, the second dose is given when half the first dose remains.) At the steady state, find the quantity of drug in the body right after a dose.
(b) If the quantity of a drug in the body after a dose is 300 mg at the steady state and if the interval between doses equals the half-life, what is the dose?
19. A dose, D, of a drug is taken at regular time intervals, and a fraction r remains after one time interval. Show that at the steady state, the quantity of the drug excreted between doses equals the dose.
20. Cephalexin is an antibiotic with a half-life in the body of 0.9 hours, taken in tablets of 250 mg every six hours.
(a) What percentage of the cephalexin in the body at the start of a six-hour period is still there at the end (assuming no tablets are taken during that time)?
(b) Write an expression for Q1, Q2, Q3, Q4, where Qn mg is the amount of cephalexin in the body right after the nth tablet is taken.
(c) Express Q3, Q4 in closed form and evaluate them.
(d) Write an expression for Qn and put it in closed form.
(e) If the patient keeps taking the tablets, use your answer to part (d) to find the quantity of cephalexin in the body in the long run, right after taking a tablet.
CHAPTER SUMMARY
- Geometric series
Finite and infinite.
- Sums of geometric series
Partial sums, convergence of infinite series.
- Applications to business and economics
Annuities, present value, multiplier effect, market stabilization.
- Applications to life sciences
Repeated drug doses, accumulation of toxins, depletion of natural resources.
REVIEW PROBLEMS FOR CHAPTER TEN
In Problems 1–8, find the sum, if it exists.
1. 2 + 22 + 23 + · · · + 210
2. 20 + 20(1.4) + 20(1.4)2 + · · · + 20(1.4)8
3. 1000 + 1000(1.08) + 1000(1.08)2 + 1000(1.08)3 + · · ·
4. 500 + 500(0.6) + 500(0.6)2 + · · · + 500(0.6)15
5. 30 + 30(0.85) + 30(0.85)2 + 30(0.85)3 + · · ·
6. 25 + 25(0.2) + 25(0.2)2 + 25(0.2)3 + · · ·
7. 
8. 
9. Around January 1, 1993, Barbra Streisand signed a contract with Sony Corporation for $2 million a year for 10 years. Suppose the first payment was made on the day of signing and that all other payments were made on the first day of the year. Suppose also that all payments were made into a bank account earning 4% a year, compounded annually.
(a) How much money was in the account
(i) On the night of December 31, 1999?
(ii) On the day the last payment was made?
(b) What was the present value of the contract on the day it was signed?
10. A drug is given in daily doses of 100 mg. After 24 hours, 82% of the previous day's dose remains in the body. What is the long-run quantity of drug in the body, right after and right before a dose is given?
11. A used car costs $15,000. The repair contract costs $500 at the end of the first year and increases by 20% at the end of each subsequent year. Find the total cost of owning the car for ten years. Include the payment at the end of the tenth year.
12. A deposit of $100,000 is made into an account paying 8% interest per year, compounded annually. Annual payments of $10,000 each, starting right after the deposit, are made out of the account. How many payments can be made before the account runs out of money?
13. This problem shows how to estimate the cumulative effect of a tax cut on a country's economy. Suppose the government proposes a tax cut totaling $100 million. We assume that all the people who have extra money spend 80% of it and save 20%. Thus, of the extra income generated by the tax cut, $100(0.8) million = $80 million is spent and becomes extra income to someone else. These people also spend 80% of their additional income, or $80(0.8) million, and so on. Calculate the total additional spending created by such a tax cut.
14. To stimulate the economy, the government gives a tax rebate totaling 5 billion dollars. Find the total additional spending resulting from this tax rebate if everyone who receives money spends
(a) 80% of it
(b) 90% of it
15. A government gives a tax rebate of N dollars to stimulate the economy. Everyone who receives money spends a fixed fraction, k, of the money received, with 0 < k < 1.
(a) Find a formula (in terms of N and k) for the total additional spending resulting from the tax rebate.
(b) If k = 0.85, what is the total additional spending as a multiple of the size, N, of the tax rebate?
16. Before World War I, the British government issued what are called consols, which pay the owner or his heirs a fixed amount of money every year forever. (Cartoonists of the time described aristocrats living off such payments as “pickled in consols.”) What should a person expect to pay for consols which pay £10 a year forever? Assume the first payment is one year from the date of purchase and that interest remains 4% per year, compounded annually. (£ denotes pounds, the British unit of currency.)
17. Figure 10.3 shows the quantity of the drug atenolol in the blood as a function of time, with the first dose at time t = 0. Atenolol is taken in 50 mg doses once a day to lower blood pressure.
(a) If the half-life of atenolol in the blood is 6.3 hours, what percentage of the atenolol present at the start of a 24-hour period is still there at the end?
(b) Find expressions for the quantities Q0, Q1, Q2, Q3, . . ., and Qn shown in Figure 10.3. Write the expression for Qn in closed form.
(c) Find expressions for the quantities P1, P2, P3, . . ., and Pn shown in Figure 10.3. Write the expression for Pn in closed form.
18. One way of valuing a company is to calculate the present value of all its future earnings. A farm expects to sell $1000 worth of Christmas trees once a year forever, with the first sale in the immediate future. What is the present value of this Christmas tree business? The interest rate is 1% per year, compounded continuously.
STRENGTHEN YOUR UNDERSTANDING
In Problems 1–30, indicate whether the statement is true or false.
1. The sum 1 + 2 + 4 + 8 + 16 + 32 is a finite geometric series with 6 terms.
2. The sum 1 + 2 + 4 + 8 + 16 + 32 + · · · + 210 is a finite geometric series with 10 terms.
3. The sum 1 + 3 + 5 + 7 + 9 + 11 is a finite geometric series with 6 terms.
4. The sum 1 − 2 + 4 − 8 + 16 − 32 is a finite geometric series with 6 terms.
5. The sum of 1 + (1/3) + (1/9) + · · · + (1/3)10 is (1 − (1/3)11)/(1 − (1/3)).
6. The sum of 3 · 1 + 3 · 2 + 3 · 4 + · · · + 3(220) is 3(1 − 220)/(1 − 2).
7. The sum of the infinite geometric series 3 + 32 + 33 + · · · +3n + · · · is 3/(1 − 3).
8. The sum of the infinite geometric series (1/2) + (1/2)2 + (1/2)3 + · · · + (1/2)n + · · · is 1/(1 − (1/2)).
9. The sum of the infinite geometric series (1/3) + (1/3)2 + (1/3)3 + · · · + (1/3)n + · · · is 1/2.
10. The sum of the infinite geometric series 5(1/2) + 5(1/2)2 + 5(1/2)3 + · · · + 5(1/2)n + · · · is 5.
11. An annuity is a sequence of equal payments or deposits made at regular intervals.
12. The present value of an annuity is the amount of money deposited today to make a series of fixed payments in the future.
13. The present value of an annuity that makes payments of $6000 per year for 10 years is less than $60,000, assuming an annual interest rate of 3%.
14. The present value of a series of constant annual payments made forever is infinite.
15. If annual deposits of $3000 are made at the beginning of each year into an account paying 2% interest per year, compounded annually, then the account balance after the fourth deposit is 3000(1.02)4 + 3000(1.02)3 + 3000(1.02)2 + 3000(1.02) dollars.
16. If annual deposits of $3000 are made at the beginning of each year into an account paying 2% interest per year, compounded annually, then the account balance after the tenth deposit is 3000(1 − (1.02)10)/(1 − 1.02) dollars.
17. A deposit of $735,000 into an account earning 5% compounded annually is sufficient to generate annual payments of $35,000 in perpetuity, starting now.
18. The present value of a payment 5 years in the future of $2000, assuming annual interest rate of 3%, is 2000(1.03)5 dollars.
19. The present value of 3 annual payments of $600, starting now, assuming annual interest rate of 4%, is 600 + 600(1.04)−1 + 600(1.04)−2 dollars.
20. The present value of 10 annual payments of $600, starting now, assuming annual interest rate of 4%, is 600(1 − (1.04)−10)/(1 − (1.04)−1) dollars.
21. If a person consumes a 25-mg tablet of a drug four times throughout the day, at the end of the day the person will have 100 mg of the drug in the body.
22. If the quantity of a drug in the body has reached the steady-state level for a person taking a drug at regular intervals, then the quantity of drug in the person's body stays constant.
23. A person consuming 3 micrograms of a toxin with breakfast each day, of which 5% leaves the body each day, has 3/(1-0.05) micrograms of toxin in the body right after breakfast, in the long run.
24. If a country used 10 billion barrels of oil last year, and oil consumption increases at 2% annually, then over the next 5 years, the total oil consumption will be 10(1.02)(1 − (1.02)5)/(1 − 1.02) billion barrels.
25. In a person taking a constant dose D mg of medication everyday, the steady-state level is the same before and after each daily dose.
26. In a person taking a constant dose D mg of medication every day, at the steady-state level the quantity of the drug eliminated daily is equal to the daily dose D.
27. If a drug is administered intravenously, then a geometric series is a better model than a differential equation for the quantity of drug in the bloodstream.
28. If a certain drug has a half-life of three hours in the body, then nine hours after a dosage of 160 mg, the quantity in the body drops to 20 mg.
29. If a 50-mg injection of a drug is given once every day, and the drug is metabolized at a continuous rate of 5% per day, then the long-term steady-state quantity in the body after injection is 50/(1 − e0.05).
30. If a drug is administered intravenously at a rate of 50 mg per day, and the drug is metabolized at a continuous rate of 5% per day, then the long-term steady-state quantity in the body is 1000 mg.
PROJECTS FOR CHAPTER TEN
1. Do You Have Any Common Ancestors?
In this project, we estimate the number of ancestors you have and determine whether you have any common ancestors. (A common ancestor is one who appears on two sides of your family tree. For example, if your great-grandmother on your mother's mother's side is also your grandmother on your father's side, then she would be a common ancestor.)
(a) In general, each person has 2 biological parents, 4 biological grandparents, 8 biological great-grandparents, and so on. Write a formula for the number of ancestors you have, going back n generations.
(b) How long is one generation? Estimate the age of typical parents when a baby is born. This is the length of time for a generation. How many generations are included if we go back 100 years? 500 years? 1000 years? 2000 years?
(c) Use your answers to parts (a) and (b) to estimate the number of ancestors you have if we go back 100 years, 500 years, 1000 years, or 2000 years.
(d) In parts (a) and (c), we counted every ancestor separately, so we assumed that you have no common ancestors. Use the fact that the population of the world was about 6 billion people in 1999 and was about 200 million people in the year 1 AD to determine whether this is a reasonable assumption. Explain your reasoning.
2. Harrod-Hicks Model of an Expanding National Economy
In an expanding national economy, the Harrod-Hicks model relates the national income in one year to the national income in the preceding year. If f(n) is the national income in year n, then the model predicts, for some constants k and h with k > 1 and h > 0, that
(a) Let C = f(0). Write f(1), f(2), and f(3) in terms of k, h, and C.
(b) Show that
Use these formulas to guess a formula for f(n).
(c) Use the formula for the sum of a finite geometric series to rewrite the formula for f(n) in closed form.
3. Probability of Winning in Sports
In certain sports, winning a game requires a lead of two points. That is, if the score is tied you have to score two points in a row to win.
(a) For some sports (e.g. tennis), a point is scored every play. Suppose your probability of scoring the next point is always p. Then, your opponent's probability of scoring the next point is always 1 − p.
(i) What is the probability that you win the next two points?
(ii) What is the probability that you and your opponent split the next two points, that is, that neither of you wins both points?
(iii) What is the probability that you split the next two points but you win the two after that?
(iv) What is the probability that you either win the next two points or split the next two and then win the next two after that?
(v) Give a formula for your probability w of winning a tied game.
(vi) Compute your probability of winning a tied game when p = 0.5; when p = 0.6; when p = 0.7; when p = 0.4. Comment on your answers.
(b) In other sports (e.g. volleyball prior to 1999), you can score a point only if it is your turn, with turns alternating until a point is scored. Suppose your probability of scoring a point when it is your turn is p, and your opponent's probability of scoring a point when it is her turn is q.
(i) Find a formula for the probability S that you are the first to score the next point, assuming it is currently your turn.
(ii) Suppose that if you score a point, the next turn is yours. Using your answers to part (a) and your formula for S, compute the probability of winning a tied game (if you need two points in a row to win).
- Assume p = 0.5 and q = 0.5 and it is your turn.
- Assume p = 0.6 and q = 0.5 and it is your turn.
4. Medical Case Study: Drug Desensitization Schedule8
Some patients have allergic reactions to critical medications for which there are no effective alternatives. In some such cases, the drug can be given safely by a process known as drug desensitization. Desensitization starts by administering a very small amount of the needed medication intravenously, and then progressively increasing the concentration or the rate that the drug is infused until the full dose is achieved.
An example of a drug desensitization regimen is shown in Table 10.1. Three solutions of the drug at different concentrations (full strength, 10-fold diluted, and 100-fold diluted) are prepared. The procedure starts with a low rate of infusion of the most dilute solution and progresses in 15-minute steps until the highest infusion rate of the most concentrated solution is reached. At the last stage, the infusion runs until the target dose is reached.
(a) For a target dose of 500 mg of a drug and a full strength solution of 2 mg/ml, make a spreadsheet that enables you to fill out the first 11 steps in Table 10.1, each of which runs for 15 minutes.
(b) How much of the drug is administered in the 12th step? How long does this step last? Fill in the last row of the table.
(c) Show how a geometric series can be used to calculate the total drug administered in the first 11 steps.
(d) How long does it take from the beginning of step 1 for the patient to receive the target dose?
1From www.usmint.gov.
2www.dallasnews.com/sharedcontent/dws/bus/stories/DN-obsoletedollarbill_22bus.State.Edition1.c8a2b4.html, accessed May 25, 2009.
3www.theoildrum.com/node/5395, accessed March 13, 2013.
4CIA World Factbook 2008.
5www.eia.doe.gov/emeu/international/oilcomsumption.html, accessed March 13, 2013.
6Statistical Review of World Energy 2010, www.bp.com.
7www.worldenergyoutlook.org/docs/weo2008/weo2008_es_english.pdf
8From David E. Sloane, M.D.