Between houses, cars, or education, most people will take out a loan at some time in their lives. In a typical loan situation, a person borrows an amount $Bo at a fixed interest rate with a fixed payback period. The borrower makes monthly payments until the loan balance (the amount that remains to be paid) is reduced to zero. The goal of this project is to set up a mathematical model that describes the balance in the loan for each month. The results of this project can be calculated and presented very effectively using a spreadsheet. Let's take a specific example. Assume you borrow Bo = $15,000 with a fixed annual interest rate of 6%, or 0.5% per month. As a first problem, assume that your monthly payment is $500. The goal is to compute the number of months required to pay off the loan. Every month, two things happen: Interest, which is 0.5% of the current balance, is added to the current balance and the loan balance is decreased by the monthly payment of $500. We let B, be the loan balance after the nth payment. B = Bo+0.005B-500 = 1.005B0-500. B=1.005" Bo-500(1+1.005 + 1.0052 +. +1.005"), where n = 0, 1, 2, .... +1.005 + 1.005 + ... + 1.005-200(1.005" - 1). 6. Substituting this expression for the geometric sum and letting Bo = $15,000, show that the loan balance after the nth payment is B = 100,000-85,000-1.005". 7. Graph the sequence of loan balances {B} for n = 0, 1, 2, ..., 40. 8. From the graph in Step 7, estimate how many months are required to reduce the loan balance to zero. 9. Use Step 6 to solve the equation B = 0. How many months are required to reduce the loan balance to zero? 10. How much interest do you pay for a loan in Steps 1-9? 11. We now consider a more realistic problem. Suppose you borrow B0 = $15,000 at a monthly interest rate of 0.5%. More typically the term of the loan (the payback period) is specified in advance and the monthly payment is determined as a function of the term. Use the argument in Steps 4-6, leaving the monthly payment m unspecified, and show that the loan balance after n months is B = 200m+(15,000 - 200m) 1.005" 12. Assume the term of the loan is 5 years (n = 60 months). Show that the monthly payment is m = $290. 13. Now let's derive the general loan amortization formula. Suppose you assume a loan of $B with a monthly interest rate of i (expressed as a decimal) and a payback period of N months. Show that your monthly iB payment is m 1-(1+i)-