25.1.1The Bad and Strange News: Divergence
25.1.2The bouncing ball
25.2
25.3
You find a snazzy red roadster, a 1994 FireBall XL, for sale at a local used car dealer.
The window price is
$5,000. The dealer offers financing with no down payment.
The salesperson explains that the car is only
$200 per month with payments for 3 years.
It sounds like a great start to the summer, because you expect to have a job that will more than cover the payment.
But wait, that's
$200 times 36 months,
You tell the salesperson, "That's seven thousand two hundred dollars!" I thought the car was only five thousand.
He pulls out his calculator and computes
Monthly computation of interest at a (nominal) annual rate r is done as follows
Problem CD-25.1
Installment Interest
Use the formula for a finite geometric series to rewrite your series as the formula
In the car payment problem, after 36 months, you owe nothing.
This makes the computation of the true interest rate a root finding problem,
Problem CD-25.2
The True Rate
Also use the computer to print a table of the remaining balance on a
$5000 loan with
$200 monthly payments that really is only charging 14% (annual) interest.
How many months does it take to pay this loan off?
25.2.1
25.2.2
. Whoa!
saying, that's only fourteen percent and you have no down payment. (Actually, his calculator says nearly fifteen percent.) You really like the car, but know the calculation smells fishy.
You would like to know the actual rate of interest r.
This is the amount you would owe at rate r (as a decimal) after one month.
At the end of the month, you pay your two hundred dollars,
At the end of the second month, you owe interest on this balance,
Then, you make your payment, leaving the outstanding balance
At the end of the third month, you owe
times this, but make another payment,
Write a formula for the balance owed on an initial debt of
$5000 borrowed at annual rate r after n monthly payments of
$200. Express your answer as a series in terms of the combined quantity
.
or
Use the FindRoot or Solve computer command with an initial guess of 16% to find the true interest rate in a five thousand dollar loan with 36 monthly payments of
$200.