In a recent post Professor Gowers writes on the new proposal for the A-level exam on the “Use of Mathematics”. In it he discusses the following discrete problem which he approaches in a way that uses continuous techniques.
” Suppose for simplicity that the interest rate for an interest-only mortgage would be 5% and that this rate never changes. If I take out a repayment mortgage of £50,000 and pay £500 a month, then roughly how long will it take me to pay off the mortgage? “
Here are two comments of mine concerning this problem.
1. “For your discrete problem about repayment mortgages I present a proposal for a direct solution. This solution, even if correct, is of course far less instructive than (*) your reasoning!
In general we have to find the value of constant payments 
Now we have to sum a geometric progression with ratio (**) 
or
For the given problem, the payments will be made during 
or
Solving for
and, as you proved
2. “Please let me only add one further interpretation of mine: the continuous effective interest rate can be derived from the 5% nominal interest rate, compounded (***) 
![\underset{m\rightarrow \infty }{\lim }\left( 1+\dfrac{0.05}{m}\right) ^{m}-1=\underset{m\rightarrow \infty }{\lim }\left[ \left( 1+\dfrac{0.05}{m}\right) ^{m/0.05}\right] ^{0.05}-1](http://s0.wp.com/latex.php?latex=%5Cunderset%7Bm%5Crightarrow+%5Cinfty+%7D%7B%5Clim+%7D%5Cleft%28+1%2B%5Cdfrac%7B0.05%7D%7Bm%7D%5Cright%29+%5E%7Bm%7D-1%3D%5Cunderset%7Bm%5Crightarrow+%5Cinfty+%7D%7B%5Clim+%7D%5Cleft%5B+%5Cleft%28+1%2B%5Cdfrac%7B0.05%7D%7Bm%7D%5Cright%29+%5E%7Bm%2F0.05%7D%5Cright%5D+%5E%7B0.05%7D-1&bg=ffffff&fg=000000&s=0 "\underset{m\rightarrow \infty }{\lim }\left( 1+\dfrac{0.05}{m}\right) ^{m}-1=\underset{m\rightarrow \infty }{\lim }\left[ \left( 1+\dfrac{0.05}{m}\right) ^{m/0.05}\right] ^{0.05}-1")
(of which approximates your formula 
![=\underset{m\rightarrow \infty }{\lim }\left[ \left( 1+\dfrac{r}{m}\right) ^{m/r}\right] ^{r}-1=e^{r}-1.](http://s0.wp.com/latex.php?latex=%3D%5Cunderset%7Bm%5Crightarrow+%5Cinfty+%7D%7B%5Clim+%7D%5Cleft%5B+%5Cleft%28+1%2B%5Cdfrac%7Br%7D%7Bm%7D%5Cright%29+%5E%7Bm%2Fr%7D%5Cright%5D+%5E%7Br%7D-1%3De%5E%7Br%7D-1.&bg=ffffff&fg=000000&s=0 "=\underset{m\rightarrow \infty }{\lim }\left[ \left( 1+\dfrac{r}{m}\right) ^{m/r}\right] ^{r}-1=e^{r}-1.")
From a point of view of a purely mathematical problem the model you are discussing in the context of the “use-of-maths A’ level “ is much more interesting and worthy.”
Corrections to the original comments:
(*) “that” in original
(**) “rate” in original
(***) “compound” in original
(****) [Edited on July 28, 2009,. In original: "which approximates your formula
[Edited on July 14, 2009: First paragraph corrected and title changed]
[Edited on July 15, 2009: Last paragraph. Remark: in a comment Professor Gowers wrote "I wasn’t suggesting that the mortgage question would be a suitable one for a use-of-maths A’level" ]
