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.

Balance after payment ⇓ a_{n} = (1 + 0.1 / 12) \cdot = (1 + 0.1 / 12) \cdot Balance before payment ⇓ a_{n - 1} - - Payment ⇓ 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

1 6^{th}

row shows that after the usual

1 5^{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.

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Recursive Rule

Spreadsheet