Right, this is an exercise in Apostol, which I am not being able to solve. I was able to prove this result for a small case, that is the case when $n=2$, $[x] + \Bigl[x + \frac{1}{2}\Bigr]=[2x]$, but I am struggling with the generalization.
Prove that $$\sum\limits_{k=0}^{n-1} \Biggl[x + \frac{k}{n}\Biggr] = [nx],$$ where $[ \ ]$ denotes the greatest integer function.
Can this be proved via induction, I ask this because I have shown it to be already true for $n=2$.