For any real number $a$ and a positive integer $n$, there is a concise formula to calculate
$$a + 2a + 3a + \cdots + na = \frac{n(n+1)}{2} a.$$
The proof for the same is given in Mathematical literature.
Is there any such formula to calculate:
$$\lfloor a\rfloor + \lfloor 2a\rfloor + \lfloor 3a\rfloor + \cdots + \lfloor na\rfloor $$
and
$$\lceil a\rceil + \lceil 2a\rceil + \lceil 3a\rceil + \cdots + \lceil na\rceil $$
for any whole number $n$ and $ 0 < a < 1$ ? Also, provide the proof for the same.