I am constantly brain storming challenge questions for my pre calculus students. Today I came up with this question, but upon finding the answer I don't think this one will be feasible for my students.
Suppose during month zero your credit card balance is $b_0$. Sometime during month zero you make the minimum payment, say $p$. The next month the bank calculates your new balance by $b_0 - p + I(b_0 - p)$ where $I$ represents your interest. Assuming you always pay the minimum payment every month, $p$, and your interest $I$ stays fixed, a recursive function that gives your balance in month $n$ could be given by $f(0) = b_0$ and $f(n) = f(n-1)-p + I(f(n-1) - p)$ for $n \geq 1$. Find a closed form for the function $n$.
As always, I set out to make sure I could even solve the problem before giving it to my students as a challenge problem. My solution took me quite a while. Here is the answer I came up with, that could possibly be further simplified:
$$f(n) = \frac{p(1+I)^{n+1} - Ib_0(1+I)^n - P(1+I)}{-I}$$
My method was to start by writing out the first few terms using the recursive definition, then start rearranging and looking for patterns. After a while I was able to spot and prove a binomial expansion, and a geometric sum of binomials was occurring which yields my solution. I will currently not write my full derivation unless requested because it is long and mostly just symbol crunching.
My question to the MSE community is, does anyone have a slick easy derivation of a solution to this problem? I did not thing it should be so tedious when I came up with it? Is there a better method?
I don't even know what to tag this, if anyone has any good suggestion for tags let me know.