If i define $f(m,n)=$ $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$
Then prove $$f(m+n,n) - f(m,n) =\frac{n^2-n}{4}$$ for all $m$ and $n$.
This question came from part of answer from this question: A sum of fractional parts.