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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 $n=2$.

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    See also: http://math.stackexchange.com/questions/184361/floor-function-properties-2x-x-x-frac12-and-nx-sum-k2016-08-17

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