Exercise 58 Page 449

a Let

a_{n}

be balance after the

n^{th}

payment. We are asked to find the balance after the fourth payment,

a_{4} .

To do this, we will find a recursive rule for

a_{n} .

Then we will use a spreadsheet.

Since the annual interest rate is

11.5 % = 0.115,

the monthly interest rate is

0.115 / 12 .

This tells us that the balance increases by a factor of

1 + 0.115 / 12

each month, and then the payment of

$ 173.86

is subtracted.

Balance after payment ⇓ a_{n} = (1 + 0.115 / 12) \cdot = (1 + 0.115 / 12) \cdot Balance before payment ⇓ a_{n - 1} - - Payment ⇓ 173.86

You borrow

$ 10000 .

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

$ 10000 .

Therefore,

a_{0} = 10000 .

Now, we can use a spreadsheet to find the balance after the fourth payment,

a_{4} .

Let's enter

a_{0} = 10000

in cell A1 and write

= Round ((1 + 0.115 / 12) * (A1) - 173.86, 2)

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

Next, copy cell A2, highlight cells A3 through A5, and paste.

We get $9683.38$ in the fifth row. Therefore $a_{4} = 9683.38,$ and this tells us that the balance after the fifth payment is $$ 9683.38 .$

b We are asked to find the balance after the last payment. You borrow the money for

7

years. This tells us that you have

7 \cdot 12 = 84

payments. Again, let's use the spreadsheet. Let's enter

a_{0} = 10000

in cell A1 and write

= Round ((1 + 0.115 / 12) * (A1) - 173.86, 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. Therefore, the balance after the usual

8 4^{th}

payment of

$ 173.86

will be in cell A85. Next, in the spreadsheet, copy cell A2, highlight cells A3 through A85, and paste.

The spreadsheet shows that after the usual

8 4^{th}

payment of

$ 173.86

the balance is

$ 0.64 .

Therefore, the last payment should be

$ 173.86 + $ 0.64 = $ 173.48 .

Communicate Your Answer p.441

Monitoring Progress p.442-446

Exercises p.447-450