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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.

  • 1
    Have you tried calculating any, looking for patterns?2012-10-05
  • 0
    Yes, tried calculating for both things. There is a pattern for particular numbers like 0.2, 0.4, 0.8. But, how to generalize the pattern for any real number like 0.39856, 0.0009843, etc? Of course, finding general patterns helps in providing a formula. There doesn't seem having a general pattern among all real numbers.2012-10-05
  • 0
    For rational $a$, your sums come up in some of the early proofs of quadratic reciprocity --- see, e.g., Eisenstein's proof http://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity2012-10-05

2 Answers 2