Exercise 57 Page 449

a Let

a_{n}

be balance after the

n^{th}

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

a_{5} .

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

9 % = 0.09,

the monthly interest rate is

\frac{0 . 0 9}{1 2} = 0.075 .

This tells us that the balance increases by a factor of

1 + 0.0075 = 1.0075

each month, and then the payment of

$ 91.37

is subtracted.

Balance after payment ⇓ a_{n} = 1.0075 \cdot = 1.0075 \cdot Balance before payment ⇓ a_{n - 1} - - Payment ⇓ 91.37

You borrow

$ 2000 .

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

$ 2000 .

Therefore,

a_{0} = 2000 .

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

a_{5} .

Spreadsheet

Let's enter

a_{0} = 5000

in cell A1 and write

= Round ((1.0075) * (A1) - 91.37, 2)

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

Next, copy cell A2, highlight cells A3 through A6, and paste.

We get a number

1612.38

in the sixth row. Therefore,

a_{5} = 1612.38,

and this tells us that the balance after the fifth payment is

$ 1612.38 .

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

$ 2000

for

2

years. Therefore, the

2 4^{th}

payment is the last one. Again, let's use the spreadsheet. Let's enter

a_{0} = 2000

in cell A1 and write

= Round ((1.0075) * (A1) - 91.37, 2)

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

Next, copy cell A2, highlight cells A3 through A26, 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

2 4^{th}

payment of

$ 91.37

will be in cell A25.

The spreadsheet shows that after the usual

2 4^{th}

payment of

$ 91.37

the balance is

$ 0.02 .

Therefore, the last payment should be

$ 91.37 + $ 0.02 = $ 91.39 .

Subchapter links

Communicate Your Answer p.441

Monitoring Progress p.442-446

Exercises p.447-450