There was a problem in Apostol's book namely, to prove that: $$\sum\limits_{k=1}^{n} \Biggl\lfloor{\frac{k}{2}\Biggr\rfloor} = \Biggl\lfloor{\frac{n^{2}}{4}\Biggr\rfloor}$$
which i could solve. The following probelm is eluding me:
- If $a =1,2, \cdots, 7$, prove that there exists an integer $b$ depending on $a$ such that $$\sum\limits_{k=1}^{n} \Biggr\lfloor{\frac{k}{a}\Biggr\rfloor} =\Biggl\lfloor{\frac{(2n+b)^{2}}{8a}\Biggr\rfloor} $$