KöMaL 2014 B4672

Determine all real functions $f$ defined on the positive integers, such that for all positive integer $n$ , $\frac{p}{f(1)+f(2)+\dots +f(n)} =\frac{p+1}{f(n)} -\frac{p+1}{f(n+1)}$, where $p$ is a fixed positive number.