Proving Hermite's identity using induction

Question

Can someone help me? This should be easy but I couldn't find it on any book or the internet. $$ \sum_{k=0}^{n-1}\left\lfloor x + \frac{k}{n}\right\rfloor = \lfloor nx \rfloor $$

Answer 1

You want to prove that $$\sum_{k=0}^{n-1}\left\lfloor x+ \frac{k}{n}\right\rfloor=\left\lfloor nx\right\rfloor$$

Question you need to ask is what should I be inducting on? Contrary to the usual perception the induction should not be on $n$ because $n$ is a "given" positive integer. The induction should be on the unique natural number $m$ for which $$\frac{m}{n} \leq x < \frac{m+1}{n}.$$

Hopefully this will help you start the induction process.