18.5 Applications of Geometric Sums and Series 581 The sum of money in the account is $200(1.05) 3 + $200(1.05) 2 + $200(1.05) + $200 = $862.03. Marietta cannot yet take her trip. On January 1, 2004, before Marietta makes a fifth payment, the first payment has grown to $200(1.05) 4 , the second payment has grown to $200(1.05) 3 , the third payment has grown to $200(1.05) 2 , the fourth payment has grown to $200(1.05). The sum of money in the account is $200(1.05) 4 + $200(1.05) 3 + $200(1.05) 2 + $200(1.05). This is just the previous sum multiplied by 1.05 (because the money has collected interest for another year). $862.03 · (1.05) = $905.12. Marietta can be on her way, with $5.12 in her pocket as a slush fund. ◆ ◆ EXAMPLE 18.12 Mass Millions is a state-run lottery encouraging residents to support the state’s public services by dangling the elusive prize of $1 million for the price of a winning lottery ticket. Instead of a million dollars, however, the winner actually receives 20 annual payments of $50,000. While this is a hefty sum, is this really a prize of $1 million? If you received $1 million today and put it into a bank account paying interest at a rate of 5% per year, just by taking the interest at the end of each year, you could pay yourself $50,000 per year starting one year hence and continuing on forever. The original million would stay in the bank generating interest. The 20 payments of $50,000 spread out over 20 years is not really the same as winning $1 million paid up front now. Assume the first payment is made to you today and the 20th payment 19 years later. Let’s compute the “up-front value” of the prize money of 20 annual payments of $50,000. The first payment of $50,000 received today is certainly worth $50,000 today. But how much is the second payment of $50,000 worth to you right now? How much is the 20th payment worth right now? Let’s rephrase the question. Let’s suppose that at the moment you win the prize, the state creates a bank account especially for you. The state puts a certain amount of money, $P , into the account today and lets it earn interest. From this account the state doles out your 20 payments of $50,000; the final payment depletes your account. How much would the state have to deposit today to make all 20 payments, each at the allotted time? This sum depends on the interest rate in the bank account. Let’s suppose that the account pays interest at 5% per year compounded annually. 15 Find P , the total amount of money in the account earmarked for you. SOLUTION Let’s begin by breaking down the problem into manageable pieces. The state must put away $50,000 for the very first payment. How much money must the state put away now in order to pay you $50,000 one year from now? We know that money in this bank account grows according to M(t) = M 0 (1.05) t .In one year we want $50,000; we must solve for M 0 . 15 Assuming a fixed 5% interest rate over a 20-year period is a rather unrealistic assumption, but we make it to simplify our model. Assume that all payments after the original are made immediately after interest is credited. 582 CHAPTER 18 Geometric Sums, Geometric Series 50,000 = M 0 (1.05) 1 M 0 = 50,000 1.05 ≈ 47, 619.05 The state must put $47,619.05 in the account in order to pay you $50,000 in one year. The quantity $47,619.05 is called the present value of $50,000 in one year at an interest rate of 5% per year compounded annually. Definition The present value of M dollars in t years at interest rate r per year compounded n times per year is the amount of money that must be put in an account (paying interest r per year compounded n times per year) now in order to have M dollars at the end of t years. If an account has an interest rate of r per year compounded n times a year, then M = M 0 1 + r n nt future value present value r n = interest per compounding period , where M is the amount of money t years in the future and M 0 is the initial deposit, the present value. In this case the present value of M dollars in t years is M 0 = M 1+ r n nt . If an account has interest rate r and interest is compounded continuously, then M = M 0 e rt and the present value of M dollars in t years is M 0 = M e rt . To continue with our lottery problem, we ask how much money the state must put away in order to pay you $50,000 in two years. (With our new vocabulary, we can rephrase this: “What is the present value of $50,000 in two years at an interest rate of 5% per year compounded annually?”) 50,000 = M 0 (1.05) 2 so M 0 = 50,000 (1.05) 2 ≈ 45,351.47 We continue in this vein. The present value of the 1st payment is $50,000. The present value of the 2nd payment is $50,000 1.05 ≈ $47,619.05. The present value of the 3rd payment is $50,000 (1.05) 2 ≈ $45,351.47. . . . The present value of the 20th payment is $50,000 (1.05) 19 ≈ $19,786.70. Notice that the amount the state must put away now in order to pay you $50,000 in 19 years is only $50,000 (1.05) 19 ≈ $19,786.70. The “up-front” value of the prize can be thought of as the sum of the present values of all of the payments. We can represent this diagramatically. Each of the 20 payments of $50,000 is being pulled to the present. 18.5 Applications of Geometric Sums and Series 583 $50,000 $50,000 $50,000 $50,000 • • • Now $50,000 + $50,000 1.05 + $50,000 (1.05) 2 +···+ $50,000 (1.05) 19 This is a geometric sum with r = 1 1.05 . We can compute the sum by putting it in closed form. P = $50,000 + $50,000 1.05 + $50,000 (1.05) 2 +···+ $50,000 (1.05) 19 1 1.05 P = $50,000 1.05 + $50,000 (1.05) 2 +···+ $50,000 (1.05) 19 + $50,000 (1.05) 20 1 − 1 1.05 P = $50,000 − $50,000 (1.05) 20 P = $50,000− $50,000 (1.05) 20 1− 1 1.05 ≈ $654, 266.04. ◆ Work through the following two examples to make sure that the notion of present value makes sense. These examples are designed to emphasize the set-up of the problems, not the summation of a geometric series. ◆ EXAMPLE 18.13 Onnie has just won an award of $1000 per year for four years, with the first of the four payments being made to him today. Suppose that the money to finance this award is being kept in a bank account with 5% interest compounded annually. How much must be in the bank right now in order to pay for his award? In other words, what is the present value of his award? SOLUTION Making Estimates. Although Onnie will receive a total of $4000 dollars, he is not getting it all right now. The bank account earmarked for this award needs less than $4000 in it because the money in the account will earn interest. Our goal is to figure out exactly what sum must be put in the account right now so that after Onnie’s award has been paid the money in the account is depleted. We expect an answer slightly less than $4000. Strategy: Treat each of the four payments separately. Calculate how much must be in the account to make the first payment, the second, the third, and the fourth. Then sum these four figures. In other words, sum the present values of the four payments. We can represent this diagramatically. We’ll pull each of the payments back to the present. $1,000 $1,000 $1,000 $1,000 We know that the present value of $1000 in t years at an interest rate of 5% per year is given by present value = $1000 1.05 t . The present value of the first payment =$1000. 584 CHAPTER 18 Geometric Sums, Geometric Series The present value of the second payment = $1000 1.05 . The present value of the third payment = $1000 1.05 2 . The present value of the fourth payment = $1000 1.05 3 . The present value of the award = the sum of the present values of the first, second, third and fourth payments. present value of award = $1000 + $1000 1.05 + $1000 1.05 2 + $1000 1.05 3 There are only four terms here, so we’ll just add them up. This is a geometric sum with a = 1000, r = 1 1.05 . present value of award = $1000 + $952.38 + $907.03 + $863.84 = $3723.25 The present value of the award is $3723.25. ◆ ◆ EXAMPLE 18.14 Julie has just won an award of $1000 per year for four years, with the first of the four payments being made to her three years from today. Suppose that the money to finance this award is being kept in a bank account with 5% interest compounded annually. How much must be in the bank right now in order to pay for her award? In other words, what is the present value of her award? SOLUTION Making Estimates. Although Julie will receive a total of $4000, she is not getting any of it right now. The bank account earmarked for this award needs less than $4000 in it because the money in the account will earn interest. Our goal is to figure out exactly what sum must be put in the account right now so that after Julie’s award has been paid the money in the account is depleted. We expect an answer substantially less than $3723.25. Strategy: Treat each of the four payments separately. How much must be in the account to make the first payment? The second? The third? The fourth? Then sum these four figures. We can represent this diagramatically. We’ll pull each of the payments back to the present. $1,000 $1,000$1,000$1,000 Now 3 years • • •• • • We know that the present value of $1000 in t years at an interest rate of 5% per year is given by present value = $1000 1.05 t . The present value of the first payment = $1000 1.05 3 . The present value of the second payment = $1000 1.05 4 . The present value of the third payment = $1000 1.05 5 . The present value of the fourth payment = $1000 1.05 6 . 18.5 Applications of Geometric Sums and Series 585 The present value of the award = the present values of the first, second, third, and fourth payments. present value of award = $1000 1.05 3 + $1000 1.05 4 + $1000 1.05 5 + $1000 1.05 6 present value of award = $863.84 + $822.70 + $783.53 + $746.22 = $3216.29 The present value of the award is $3216.29. ◆ ◆ EXAMPLE 18.15 Suppose a philanthropic organization wants to start a fund that will make payments of $2000 each year to the American Cancer Society. The payments are to begin in five years and go on indefinitely. (They are setting up what is known as a perpetual annuity.) The fund will be kept in an account with a guaranteed 6% annual interest compounded continuously. How much money should be put in the fund today so the payments can begin five years from today? SOLUTION As is the case with many problems, there are several different constructive approaches to obtaining a solution. We’ll look at two of these. Approach 1. Our strategy is to figure out how much the organization must put away now in order to make the first payment of $2000, the second payment, and so on. We’ll then add up these figures to find the total amount that must be invested today. In other words, we will look at the sum of the present values of the payments. We can represent this diagramatically. $2,000 $2,000 $2,000 $2,000 Now • • • • • • 5 years Interest is 6% compounded continuously, so M(t) = M 0 e .06t . Solving for M 0 and denoting the future value by M gives M 0 = M e .06t . The present value of the 1st payment = $2000 e .06(5) ≈ $1481.64. The present value of the 2nd payment = $2000 e .06(6) ≈ $1395.35. The present value of the 3rd payment = $2000 e .06(7) ≈ $1314.09. The amount of money the foundation must put in the account is $2000 e .06(5) + $2000 e .06(6) + $2000 e .06(7) +···. This is an infinite geometric series with a = $2000 e .06(5) and r = 1 e .06 . |r| < 1, so the series converges to a 1−r . a 1 − r = $2000 e .06(5) 1 − 1 e .06 = $2000e −.06(5) 1 − e −.06 ≈ $25,442.17 The organization must use $25,442.17 to set up this perpetuity. 586 CHAPTER 18 Geometric Sums, Geometric Series Approach 2. (This approach does not use geometric series.) Observe that in four years there should be enough money in the bank so that in each subsequent year the interest alone is $2000. Let F be the amount of money needed in four years. By the end of the next year this sum of F dollars ought to have earned $2000 in interest. Fe .06 − F = 2000 F(e .06 − 1) = 2000 F = 2000 e .06 − 1 After four years F dollars are needed. If we let M 0 be the amount of money that must be put in the account now, then 2000 e .06 − 1 = M 0 e .06(4) M 0 = 2000 (e .06 − 1)e .06(4) , which is, to the nearest penny, $25,442.17. ◆ Time and Money From an economic standpoint, the essence of this whole discussion is that $1000 now, $1000 in 10 years, and $1000 in 20 years are not equivalent if the money could be earning interest. The $1000 now would grow to be much more than $1000 in 20 years if it were kept in an account paying interest. We can compare these three amounts by converting them to a common time frame. For instance, we can look at how much they are all worth in 20 years. If interest is 5% per year compounded monthly, then $1000 now is worth $2712.64 in 20 years. $1000 in 10 years is worth $1647.01 in 20 years. $1000 in 20 years is worth $1000.00 in 20 years. Alternatively, we can compare them in the present. This is what calculating present value does. $1000 now is worth $1000.00 now. $1000 in 10 years is worth $607.16 now. $1000 in 20 years is worth $368.64 now. REMARK If money were kept under a pillow instead of being kept in a bank paying interest, the present, future, and past values of $1000 would be the same in the context of our discussion. In this entire discussion we have concerned ourselves exclusively with interest, not inflation, the strength of the dollar in the world market, or other considerations. (Economists will often include other considerations.) 18.5 Applications of Geometric Sums and Series 587 PROBLEMS FOR SECTION 18.5 1. You have the choice of two awards. Award 1: You will receive six yearly payments of $10,000, the first payment being made three years from today. Award 2: You will receive three payments of $20,000, the payments being made at two-year intervals, the first payment being made two years from today. Suppose that the interest rate at the bank is 4% per year compounded quarterly. (a) Find the present value of award 1 and the present value of award 2. Which present value is larger? Which award scheme would you choose? (b) Suppose you put each payment in the bank as soon as you receive it. How much money will be in the account eight years from today under the first award scheme? Under the second award scheme? 2. Banking Basics: We’ve been looking at banking because it ties in with the theoretical mathematics we are studying and because most of us have some sort of interface with a bank, be it only via an ATM card. The following questions are designed to direct your attention to the basics. (a) Suppose a bank compounds interest n times per year, n>1. Which will be larger, the nominal interest rate, or the effective interest rate? (b) Suppose interest rates are fixed at r % per year compounded annually. Which is larger, the present value of $1000 in T years or $1000 in (T + 1) years? (c) Which is larger, the present value of $1000 in T years at a rate of 4% compounded annually or the present value of $1000 in T years at a rate of 5% compounded annually? 3. (a) A friendly benefactor, impressed with Joselyn’s enthusiasm for her mathematical studies, decides to award her a scholarship of $6000 to be paid to her five years from today. How much money must the benefactor put aside today in an account earning a nominal annual interest of 4% compounded continuously in order to cover Joselyn’s award? (This question asks “what is the present value of $6000 in five years at an interest rate of 4% compounded continuously?”) (b) Another benefactor, interested in Patrick’s potential, promises Pat that if he contin- ues his studies in mathematics he will be awarded a scholarship. The scholarship will given in three payments of $2000, the first payment being made in three years (when he graduates), the second in four years, and the last payment being made five years from today. The benefactor has put aside money for Pat’s scholarship in an account earning 4% nominal annual interest compounded continuously in order to cover Pat’s award. Pat says that he will promise to take mathematics, but he would like his scholarship money up front and immediately. Because the bene- factor has not set aside the full $6000 now, he agrees to give Pat the present value of the award, i.e., the amount of money he has set aside in an account for Pat. How much money should Pat be expecting? (Hint: You need to do three separate calculations. Find the present value of the first payment, then of the second, and then of the third.) 588 CHAPTER 18 Geometric Sums, Geometric Series (c) Which answer did you expect to be bigger, the answer to part (a) or the answer to part (b)? Why? Have your calculations matched your expectations? 4. A ball is thrown from the ground to a height of 16 feet. Each time the ball bounces it rises up to 60% of its previous height. What is the total distance traveled by the ball? (Hint: Keep in mind that between every bounce the ball is going up and then coming back down.) In Problems 5 through 12, make initial estimates to be sure that the answers you get are in the right ballpark. 5. Suppose you borrow $10,000 at an interest rate of 7% compounded annually. You begin paying back money one year from today and make uniform payments of $P annually. You pay back the entire debt after 10 payments. What are your annual payments? Hint: Pull each of the 10 payments of $P back to the present. The sum should be $10,000. Ballpark figures: If no interest were charged, then you would pay $10,000/10 = $1000. If you had to pay interest on the entire $10,000 for 10 years, then you’d pay $10,000 ·1.07 10 /10. The actual answer is somewhere between these two extremes. 6. Suppose you borrow $10,000 at an interest rate of 6% compounded annually. You begin paying back the loan one year from today and make uniform payments annually. You pay back the entire debt after 8 payments. What are your annual payments? 7. Suppose you borrow $10,000 at an interest rate of 7% compounded annually. You begin paying back money five years from today and make payments annually. You pay back the entire debt after 10 payments. What are your fixed annual payments? 8. Suppose you borrow some money at an interest rate of 6% compounded annually. You begin paying back one year from today and make payments annually. You pay back the entire debt after 10 payments of $1000 each. How much money did you borrow? 9. Suppose you borrow some money at an interest rate of 6% compounded monthly. You begin paying back one month from today and make payments monthly. You pay back the entire debt after 180 payments of $1000 each. (This is a 15-year mortgage.) How much money did you borrow? 10. Suppose you borrow some money at an interest rate of 6% compounded monthly. You begin paying back one year from today and make payments annually. You pay back the entire debt after 30 payments of $1000 each. How much money did you borrow? 11. Suppose you are saving for a big trip abroad. You estimate that you’ll need $4000. You plan to put away a fixed amount of money every month for the next two years (24 deposits) so that immediately after the 24th deposit you have enough money for your trip. You put your money into an account paying interest of 4.5% per year compounded monthly. How much must you deposit every month? 12. Suppose you are saving to buy some cattle. You plan to put $200 into an account every month for the next three years (36 deposits) to pay for the cows. You put your money into an account paying interest of 4.5% per year compounded monthly. Immediately after the 36th deposit, how much money will you have in your cattle fund? 18.5 Applications of Geometric Sums and Series 589 13. People who have slow metabolism due to a malfunctioning thyroid can take thyroid medication to alleviate their condition. For example, the boxer Muhammad Ali took Thyrolar 3, which is 3 grains of thyroid medication, every day. The amount of the drug in the bloodstream decays exponentially with time. The half-life of Thyrolar is 1 week. (a) Suppose one 3-grain pill of Thyrolar is taken. Write an equation for the amount of the drug in the bloodstream t days after it has been taken. (Hint: In part (a) you are dealing with one 3-grain pill of Thyrolar. Knowing the half-life of Thyrolar, you are asked to come up with a decay equation. This part of the problem has nothing to do with geometric sums.) (b) Suppose that Ali starts with none of the drug in his bloodstream. If he takes 3 grains of Thyrolar every day for five days, how much Thyrolar is in his bloodstream immediately after having taken the fifth pill? (c) Suppose Ali takes 3 grains of Thyrolar each day for one month. How much thyroid medication will be in his bloodstream right before he takes his 31st pill? Right after? (d) After taking this medicine for many years, what was the amount of the drug in his body immediately after taking a pill? Historical note: Before one of his last fights Muhammad Ali decided to up his dosage to 6 grains. In doing so he mimicked the symptoms of an overactive thyroid. The result in terms of the fight was dismal. 14. Amanda, at the young age of 9, has gotten it firmly into her mind that she wants to be a doctor when she grows up. Her father, panic-stricken, wonders how the family will finance her college and medical school education. To assuage his anxiety he decides to set aside enough money in a bank account right now so that they will be able to withdraw $10,000 every year for eight years beginning nine years from today. Amanda’s mother computes how much money they will have to put into the account with an annual interest rate of 6% compounded quarterly. What figure should she arrive at? 15. You have found a calling! You have some burning questions about elephants and want desperately to go to Kenya for a year. In addition to the plane fare you’ll need some equipment, a guide, a jeep You ’ll need some money. You figure that you’ll need $7000. Each month beginning today you plan to put a fixed amount of money into an account paying 6% interest compounded monthly. How much must you deposit into the account each month if you plan to begin your field work in four years? 16. A woman takes out a loan of $100,000 in order to finance a home. The interest rate is 12% per year compounded monthly and she has a 30-year mortgage. She will pay back the loan by paying a fixed amount, M dollars, every month beginning one month from today and continuing for the next 30 years. (a) What is M?(Hint: The sum of the present values of her 360 payments, pulled back to the present using an interest rate of 12%, should equal her loan.) (b) How much could she save each month if she could borrow at an interest rate of 6.75% per year compounded monthly? 17. Mike L. and Mike C. have decided to establish the Mike and Mike Math Millenium Miracle Prize. The M&M M 3 prize is worth $2000 to the lucky winner. Due to limited 590 CHAPTER 18 Geometric Sums, Geometric Series funds, Mike and Mike have decided to award the prize once every 4 years, starting 10 years from now and going on indefinitely. (It’s like the Fields Medal in Math, only more accessible.) They have begun to go door-to-door to take collections in order to establish the fund. How much money should the M&M M 3 Prize Fund contain right now in order to start payments 10 years from today? Assume a guaranteed interest rate of 5% per year compounded annually. 18. Barry is thinking about buying a vehicle. He hears on the radio that he can buy a truck with no money down for two years and then make monthly payments of $150. He thinks this sounds good. He asks Angie if he should buy it. Angie says that she thinks he needs to know about the interest rates and how many years he’ll have to make the monthly payments. Barry listens to the radio again and discovers that the monthly payments must be made for 10 years. He decides to compute the present value of the truck payments using an interest rate of 6% per year compounded monthly—or 0.5% per month. What answer should he get? 19. Brent and Rob were working on their math homework when Rob got a headache. Because Rob was incapacitated, Brent went to take a nap. Due to the headache he is blaming on the homework, Rob takes two aspirin. In the body aspirin metabolizes into salicylic acid, which has a half-life of two to three hours. (Source: The pharmacist at a CVS Pharmacy.) Rob is a big fellow, so for the purposes of this problem we’ll say three hours is the half-life of salicylic acid. (a) The math headache is haunting him, so three hours later Rob takes two more aspirin. In fact, the headache is so bad that every three hours he takes two more aspirin. If he keeps this up indefinitely, will the level of salicylic acid in his body ever reach the level equivalent to taking four aspirin all at once? (b) Brent wakes up from a deep sleep, looks at his math homework, gets a headache, looks at Rob, and decides that he’s going to take two aspirin every two hours. If he keeps this up indefinitely, will the level of salicylic acid in his body every reach the level equivalent to taking three aspirin all at once? Four aspirin all at once? Five aspirin all at once? (Assume again that the half-life of salicylic acid is three hours.) 20. Matt is saving money for his wedding. Suppose that at the beginning of every month he puts $300 in his savings account. The savings account gives interest of 0.5% every month, for a nominal annual interest rate of 6% per year compounded monthly. Matt does this for three years. How much will be in his savings account right after he makes the 36th deposit? 21. Nadia is saving for a trip to Venezuela. She estimates that she’ll need $3000. She plans to put away a fixed amount of money every month for the next 30 months so that immediately after the 30th deposit she will have enough money for her trip. She puts her money into an account paying interest of 4% per year compounded monthly. How much must she deposit every month? Before you begin calculations, do an estimate. Will she have to put aside more than $100 each month, or less? . mimicked the symptoms of an overactive thyroid. The result in terms of the fight was dismal. 14. Amanda, at the young age of 9, has gotten it firmly into her mind that she wants to be a doctor when she. value of the prize can be thought of as the sum of the present values of all of the payments. We can represent this diagramatically. Each of the 20 payments of $50,000 is being pulled to the. decides to award her a scholarship of $6000 to be paid to her five years from today. How much money must the benefactor put aside today in an account earning a nominal annual interest of 4% compounded