This preview shows pages 1–2. Sign up to view the full content.
CMSC16100
Lecture 2
Let's do something a bit more interesting.
..
Let's suppose you want to compute the monthly payment on a loan  a house loan,
a car loan, or perhaps even a student loan :).
There are a lot of ways we
might approach this computationally, but we're going to do this the old
fashioned way.
We're going to think about it.
..
What we want to do is to write a function
payment years rateInPercent prinicipal
Here we assume a standard loan, in which interest is compounded monthly.
Let's try to discover an expression for the balance (b_n) after making the nth
payment, in terms of the monthly payment (m), the principal (p), and the
interest rate (i).
b_0 = p
b_1 = p (1 + i/1200)  m
b_2 = p (1 + i/1200)^2  m (1 + i/1200)  m
OK. The first thing we notice is that the term (1 + i/1200) occurs a lot.
This
is basically the result of converting the interest from percentage to
unitbased, and from an incremental rate to a multiplier.
So let's let
j = (1 + i/1200) be the monthly interest multiplier.
b_0 = p
b_1 = p j  m
b_2 = p j^2  m j  m
b_3 = p j^3  m j^2  m j  m
= p j^3  m (j^2 + j + 1)
Now, at this point, we might recognize that m is being multiplied by a geometric
series, and recall the identity
j^k + j^(k1) + .
.. + j^2 + j + 1 = (j^(k+1)  1)/(j1)
Note that this requires j != 1, i.e., a nonzero interest rate. So we might
*conjecture*
b_n = p j^n  m (j^n  1) / (j1)
Now, when I was in college, induction from the first few examples was viewed as
satisfactory for elementary education majors.
I certainly don't mean to
denegrate elementary education majors, but our mathematical standards are
higher.
Can we prove this?
Yes, by induction.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/10/2009 for the course CMSC 16200 taught by Professor Kurtz during the Fall '09 term at UChicago.
 Fall '09
 Kurtz

Click to edit the document details