MATH 2305
Spring 2003
Project

Option I.

The purpose of this project is to investigate the repayment of a loan by means of a recursive model. We will consider the balance B (in dollars) of the loan over time, t (in months), and look at the rate of change of B,. The balance B can is defined recursively. The original balance, B(0), is the original amount of the loan. For t > 0, B(t) = B(t - 1) +

To work on this project you will need to collect the following information about an item to be purchased on credit:

  1. the item,
  2. the purchase price (including taxes and other costs),
  3. the down payment,
  4. the monthly payment you can afford or think is reasonable,
  5. the term of the loan, or the number of payments, and
  6. the interest rate you will pay.

If necessary, you can make reasonable guesses as to those amounts

The item could be a car or truck, a major appliance or a house, but does not need to be one of these. Do not get an estimate for the actual monthly payment from a salesperson or finance officer. In particular, do not use payment information from loans you have obtained in the past. You will determine what payment is required in the course of the lab.

The original balance, B0, will be the purchase price less the down payment. During the first as well as subsequent months, the balance will increase by the amount of the interest and decrease by one monthly payment. If r is the annual interest rate and P is the monthly payment, then at time t - 1,

= (r/12)*B(t - 1) - P.

Given the interest rate and the balance that you start with, what payments will have the rate, < 0? You should be able to explain why it is important for to be less than 0.

In class you will get a formula for finding the balance in terms of t, B0 , and P. First, use the data you collected on your purchase to determine if your guess of the payment was correct. If the loan is to be repaid at time n, the balance, B(n) should be $0. Solve the given formula for P to find a payment that makes the balance $0 at the end of the term.

As part of the project, you will need to need to prove the formula by induction.

What to turn in: A 1-3 page typed report including

  1. your source of information for the price, down payment, original choice of payment, term of loan, and interest rate,
  2. discussion about being less than 0,
  3. the resulting balance with your "guessed" payment,
  4. your solution for the monthly payment zeroing the balance, and
  5. your induction proof of the formula.

Groups: You are welcome to work with up to 2 other people on this project. If you work with others, each person should include a statement about their contributions to the project and how much they thought the other persons contributed to the project. All people working together on the project will get the same grade, the grade assigned to the single report turned in.

A set of grading criteria is below

For 2 points each, did you...

  1. Report on how you got your price, down payment and interest rate?
  2. Report a first guess on your payment and the resulting balance?
  3. Explain why the rate of change of B had to be less than 0?
  4. Correctly find the payment using the given formula?
  5. Include the basic step of the induction proof?
  6. Include the induction step of the proof?
  7. Present a clearly written report?
  8. Explain group contributions?
  9. Turn the paper in?
  10. Turn the paper in on time?

Option II:

Complete a similar problem from the exercises not assigned, or programming exercises from the end of the chapters. The report should be of similar nature to the one specified above. The topic must be cleared with the instructor by April 18.

Due Date: Tuesday, April 29 in class. It may be turned in early. Late papers will not be accepted.