You borrow $150,000 at 9% interest compounded monthly for 30 yearsto buy a new h

Question

You borrow $150,000 at 9% interest compounded monthly for 30 yearsto buy a new home. The monthly mortgage payment has been determinedto be $1206.94
(a) Find a recursive sequence that gives the balancebn of the mortgage remaining after each monthlypayment n has been made. (b) Use the table feature of a graphing utility to find thebalance remaining for every five years where 0 < n <360 (c) What is the total amount paid for a $150,000 loan underthese conditions? Explain your answer (d) How much interest will be paid over the life of theloan?
(a) Find a recursive sequence that gives the balancebn of the mortgage remaining after each monthlypayment n has been made. (b) Use the table feature of a graphing utility to find thebalance remaining for every five years where 0 < n <360 (c) What is the total amount paid for a $150,000 loan underthese conditions? Explain your answer (d) How much interest will be paid over the life of theloan?

Explanation / Answer

The question begins as if you are to set up a standard InterestCalculating Equation: I = P(1 +r/n)ntWhere: I = Interestcalculated                   r = InterestRate                                                     t = time (in years)             P = Original LoanAmount            n= number of compounds (monthly = 12) The catch here is that a mortgage payment is being applied to thisnumber (meaning that we have to subtract the mortgage payment eachmonth). So, instead of having an exponential formula, likethe one above, we'll have a recursive formula (a formula thatrequires the previous "month's" balance in order to determine theremaining balance). Month 1: We take the above formula (with out the exponentsand subtract the mortgage payment)                        New Balance = (150,000)(1 + .09/12) -1206.94                                             ˜ $149,918.06 Month 2: We take the previous balance ($149,918.06) and replace theoriginal (150,000) since this is our new balance on the loan.                         New Balance = (149,918.06)(1 + .09/12) -1206.94                                              ˜ $149,835.51 Month 3: Repeat the previous step with the new balance:                          New Balance = (149,835.51)(1 + .09/12) -1206.94                                                ˜ $149,752.33 Notice the each month, we have to replace the Original Balance withthe New Balance from the month before. This is what makesthis problem recursive.                                  bn+1 = bn(1 + .09/12) -1206.94            bn+1 represents the upcoming or current month, whilebn represents the balance from the previous month.

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