How do I prove that $[\frac{x}{n}]+[\frac{x+1}{n}]+[\frac{x+2}{n}]....+[\frac{x+n-1}{n}]=[x]$

Question

How do I prove that $[\frac{x}{n}]+[\frac{x+1}{n}]+[\frac{x+2}{n}]....+[\frac{x+n-1}{n}]=[x]$

How to start?

thanks

Answer 1

HINT

Use the fact that $[t]$ is the only integer such that $t - 1 \lt [t] \le t $

$\frac{x}{n} -1 +\frac{x+1}{n} - 1+\frac{x+2}{n} - 1…+\frac{x+n-1}{n} -1 \lt [\frac{x}{n}]+[\frac{x+1}{n}]+[\frac{x+2}{n}]…+[\frac{x+n-1}{n}] \le \frac{x}{n} + \frac{x+1}{n}+\frac{x+2}{n}…+\frac{x+n-1}{n}$

UPDATE

The OP question can be proved using Hermite's identity, by taking $x := \frac x n$ My hint was not useful at all, so please remove the acceptance.