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Suppose you take out a mortgage for

`A`

dollars at a monthly interest rate

`I`

and a monthly payment

`P`

(To calculate

`I`

: if the annual interest rate is

`12%`

, divide by 12 to get a monthly rate of

`1%`

, then replace the percentage with the decimal fraction 0.01.) Let

`A_(n)`

denote the amount you have left to pay off after

`n`

months. So,

`A_(0)=A`

by definition. At the end of each month, you are first charged interest on all the money you owed during the month, and then your payment is subtracted. So,

`A_(n+1)=A_(n)(1+I)-P`

Prove by induction that

`A_(n)=(A-(P)/(I))(1+I)^(n)+(P)/(I)`