Since Chapter 1, you have solved problems about monthly payments, such as auto loans and home mortgages. You've learned how to find the monthly payment, P, required to pay off an initial amount, A0, over n months with a monthly percentage rate, r. With series, you can find an explicit formula to calculate P. The recursive rule An = An-1 (1 + r) − P creates a sequence of the unpaid balances after the nth payment. The expanded equations for the unpaid balances are
A1 = A0 (1 + r) − P
A2 = A0 (1 + r)2 − P(1 + r) − P
A3 = A0 (1 + r)3 − P(1 + r)2 − P(1 + r) − P
and so on. Find the expanded equation for the last unpaid balance, An. Look at this equation for a partial sum of a geometric series, and use the explicit formula, Sn = u1(1-rn)1 - r, to simplify the equation for An. Then, substitute 0 for An (because after the last payment, the loan balance should be zero) and solve for P. This gives you an explicit formula for P in terms of A0, n, and r. Test your explicit formula by solving these problems.
a. What monthly payment is required for a 60-month auto loan of $11,000 at an annual interest rate of 4.9% compounded monthly? (Answer: $207.08)
b. What is the maximum home mortgage for which Tina Fetzer can qualify if she can only afford a monthly payment of $620? Assume the annual interest rate is fixed at 7.5%, compounded monthly, and that the loan term is 30 years.