Exercise 65 Page 450

Let a_n be balance after the n^(th) payment. We are asked to find how long it takes to pay back the loan. To do this, we will find a recursive rule for a_n. Then we will use a spreadsheet.

Recursive Rule

Since the annual interest rate is 10 %=0.1, the monthly interest rate is 0.1/12. This tells us that the balance increases by a factor of (1+0.1/12) each month, and then the payment of $ 213.59 is subtracted. ccccc c Balance after payment & =(1+0.1/12) * & c Balance before payment & - & Payment ⇓ & & ⇓ & & ⇓ a_n & =(1+0.1/12) * & a_(n-1) & - & 213.59

You borrow $3000. This tells us the balance at the beginning of the first month is $3000. Therefore, a_0=3000. We are asked to find how long it takes to pay back the loan. To do this we will use a spreadsheet.

Spreadsheet

Let's enter a_0=2000 in cell A1 and write =Round((1+0.1/12)*(A1)-213.59,2) in cell A2. When we hit enter we will get the following.

Notice that the balance after the first payment is in cell A2. This tells us that the balance after the n^(th) payment is in the (n+1)^(th) row. Next, in the spreadsheet, copy cell A2, highlight cells A3 through A17, and paste.

The spreadsheet in the 16^(th) row shows that after the usual 15^(th) payment of $ 213.59 the balance is $ 0.01. Therefore, the last payment should be $ 213.59+$ 0.01=$213.60. This tells us it will take 15 months to pay back the loan.