Question 58
Let $N$ denote the positive integers. Find all functions $f:N\to N$ such that, for every positive integer $n$,
$$\frac1{f(1)f(2)}+\frac1{f(2)f(3)}+\cdots+\frac1{f(n)f(n+1)}=\frac{f(f(n))}{f(n+1)}.$$
$$\frac1{f(1)f(2)}+\frac1{f(2)f(3)}+\cdots+\frac1{f(n)f(n+1)}=\frac{f(f(n))}{f(n+1)}.$$