Exercise 14 Page 446

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

7.5 % = 0.075,

the monthly interest rate is

0.075 / 12 = 0.00625 .

This tells us that the balance increases by a factor of

1 + 0.00625 = 1.00625

each month. At the end of the month, the payment of

$ 1048.82

is subtracted.

Balance after payment ⇓ a_{n} = 1.00625 \cdot = 1.00625 \cdot Balance before payment ⇓ a_{n - 1} - - Payment ⇓ 1048.82

You borrow

$ 150000

for

30

years. This tells us the balance at the beginning of the first month is

$ 150000 .

Therefore,

a_{0} = 150000 .

We are asked to find the amount of the last,

36 0^{th},

payment. To do this, we will use a spreadsheet.

Spreadsheet

Let's enter

a_{0} = 150000

in cell A1 and write

= Round ((1.00625) * (A1) - 1048.82, 2)

in cell A2. When we hit enter we will get the following.

Next, copy cell A2, highlight cells A3 through A361, and paste. 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. Therefore, the balance after the usual

36 0^{th}

payment of

$ 1048.82

will be in cell A361.

The spreadsheet in the

36 1^{th}

row shows that after the usual

36 0^{th}

payment of

$ 1048.82

the balance is

$ 2.21 .

Therefore, the last payment should be

$ 1048.82 + $ 2.21 = $ 1051.03 .

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Recursive Rule

Spreadsheet