King Tin writes the first n perfect squares on the royal chalkboard, but he omits the first (so for n=3, he writes 4 and 9). His son, Prince Tin, comes along and repeats the following process until only one number remains:
He erases the two greatest numbers still on the board, calls them a and b, and writes the value of \frac{a b-1}{a+b-2} on the board.
Let S(n) be the last number that Prince Tin writes on the board. Let
\lim _{n \rightarrow \infty} S(n)=r
meaning that r is the unique number such that for every \epsilon>0 there exists a positive integer N so that |S(n)-r|<\epsilon for all n>N. If r can be written in simplest form as \frac{m}{n}, find m+n.