Hello, person!

This confusion has turned up from time to time.

I don't knowwhois teaching that a time-payment plan is asimplegeometric series.

It is very annoying and I wish they would stop!

A mortgage is taken out for $80,000.

It is to be paid by annual instalments of $5000 with the first payment being made

at the end of the first year that the mortgage was taken out.

Interest of 4% is then charged on any outstanding debt.

Find the total time taken to pay off the mortgage.

HINT: Find an expression for the debt remaining after n years

and solve using the fact that if it is paid off, the debt = 0.

This is anAmortizationproblem.

You can (1) derive the Amortization formula, or (2) memorize the formula.

. . Neither task is pleasant.

Let: .

We borrow dollars at time zero.

In one year, they charge r% interest.

At the end of year-1, we owe: . dollars.

Then we pay dollars.

. . Our balance is: . dollars.

During year-2, they charge r% interest.

At the end of year-2, we owe: . dollars.

Then we pay dollars.

. . Our balance is: . dollars.

During year-3, they charge r% interest.

At the end of year-3, we owe: . dollars.

Then we pay dollars.

. . Our balance is: . dollars.

See the pattern?

At the end of year- , our balance is:

. . dollars.

But by year- , we expect to pay off the loan; the balance is $0.

So, we have: .

. . . . .[1]

The geometric series has the sum: .

Then [1] becomes: .

Therefore: . . . . the Amortization Formula !

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Back to the problem . . .

We are given: .

Substitute into the Formula: .

. . . .

. . . .

Take logs: .

Therefore: . years.

I need a nap . . .

.