Geometric Series

Geometric series have many important applications. In this file we use geometric series to investigate state lotteries but you may want to look at one of the other applications below.

	The nucleii of a radioactive isotope decay randomly. As a result the number of nucleii that decay in a certain period of time can be described by a geometric series. In this application we use geometric series to look at radioactive decay.
	When government or a private company creates new jobs in an area, these new jobs create additional secondary jobs. The secondary jobs lead, in turn, to additional jobs, and so forth, creating a geometric series of new jobs.
	Automobile dealers are pushing leases. These appear to be an easy way to "buy" a car -- but, in fact, you are not really "buying" the car because at the end of the lease you have little or no equity. This module application uses geometric series to compare leasing with the more traditional car loan.
	Geometric series can help us analyze the relative benefits of two different strategies for tennis and similar games -- an aggressive strategy and a more conservative strategy.

We've all seen the signs -- The $1,000,000 Prize -- but how many of us read the fine print -- $50,000 a year for 20 years. Well, that's still a million dollars -- right? -- Wrong!! If you won a real $1,000,000 you could invest it -- a reasonably safe investment might pay 7% interest or more and that would earn you $70,000 per year, not just for twenty years but forever. So $1,000,000 really is quite bit more than $50,000 per year for twenty years.

As everyone knows $50,000 this year is worth more than $50,000 ten years from now for a variety of reasons.

• If you have $50,000 right now and you don't want to spend it right away then you can invest it. A reasonably safe investment might pay 7% interest. Thus, your $50,000 this year would be worth 1.07 * $50,000 = $53,500 next year, 1.07 * $53,500 = $57,245 the following year and $98,357.57 after ten years.

• If you do need to spend $50,000 this year and want to borrow the money and pay it back ten years from now then because banks and other lenders charge interest you would need considerably more than $50,000 in ten years to pay off the loan.

• Because of inflation $50,000 this year buys considerably less than $50,000 did ten years ago. In all likelihood $50,000 will buy even less ten years from now.

1. If you borrow $50,000 from a lender that charges 12% interest and pay the loan off in one year how much will you have to pay the lender? answer

2. If you borrow $50,000 from a lender that charges 12% interest and pay the loan off in ten years how much will you have to pay the lender? answer

3. If you know that you will receive $50,000 in ten years and want to borrow as much as possible now, planning to pay the loan off when you receive the $50,000 in ten years, how much can you borrow now if the lender charges 12% interest? answer

Bankers, businesspeople, and consumers need to be able to work with income and expenditures at various different times. For example, a retail store buys stock now and sells it in the future. If it sells an item for 10% more than it paid that sounds like a 10% profit but if they took out a loan to buy the item then interest on the loan may eat away some or all of the profit.

The key idea we use in these kinds of situation is present value. The present value of an amount A of money paid n years from now is typically A / (R^n) where R is a factor that might be, for example, 1.07 or 1.12. The value of the factor R depends on circumstances. For example, someone with lots of money invested at an average rate of 8% would use R = 1.08 or someone who is in debt and paying an average rate of 15% interest would use R = 1.15.

1. Using R = 1.08 what is the present value of $50,000 paid one year from today? answer

2. Using R = 1.08 what is the present value of $50,000 paid two years from today? answer

3. Using R = 1.08 what is the present value of the $1,000,000 lottery prize -- that is, $50,000 paid today plus $50,000 paid one year from today plus, ... plus $50,000 paid 19 years from today. answer

4. Using R = 1.08 what is the present value of $50,000 per year forever -- that is, $50,000 paid today plus $50,000 paid one year from today plus, ... plus ... answer

5. Using R = 1.06 what is the present value of $25,000 earned two years ago? answer

6. Using R = 1.06 what is the present value of $25,000 earned ten years ago? answer

To find the present value of the $1,000,000 lottery prize we must compute the sum

Theorem

Use this new formula to verify your earlier computaion for the present value of the $1,000,000 lottery prize -- that is, $50,000 a year for twenty years -- using R = 1.08. Notice that since R = 1.08, r = 1/1.08.

To compute the present value of $50,000 per year forever we need to compute the sum

Check Your Understanding

1. Find the present value of $50,000 per year for life using R = 1.08 and the formula above.

2. Find the present value of $50,000 per year for life using R = 1.08 as follows. First, find out how much you would have to invest today to earn $50,000 interest per year. This investment takes care of every payment except the first payment. Add $50,000 to determine the present value of $50,000 per year for life. Note: Your answer for this problem should agree with your answer for the previous problem.

You may want to look at one of the other applications below.

	The nucleii of a radioactive isotope decay randomly. As a result the number of nucleii that decay in a certain period of time can be described by a geometric series. In this application we use geometric series to look at radioactive decay.
	When government or a private company creates new jobs in an area, these new jobs create additional secondary jobs. The secondary jobs lead, in turn, to additional jobs, and so forth, creating a geometric series of new jobs.
	Automobile dealers are pushing leases. These appear to be an easy way to "buy" a car -- but, in fact, you are not really "buying" the car because at the end of the lease you have little or no equity. This module application uses geometric series to compare leasing with the more traditional car loan.
	Geometric series can help us analyze the relative benefiots of two different strategies for tennis and similar games -- an aggressive strategy and a more conservative strategy.