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

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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)

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