Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm trying to construct a proof that for any odd integer: the ceiling of $\large \lceil \frac{N^2}{4} \rceil = \frac{N^2 + 3}{4}$.

Anyone have a second to show me how this is done? Thanks!

share|cite|improve this question
Pleas do not deface your questions. This devalues the good answers it has received. – robjohn May 6 '13 at 20:07

Hint: what is $N^2 \pmod 4$ for odd $N$?

share|cite|improve this answer

Let $N=2k+1$. Then the LHS term is.... The RHS term is.... Hence they are equal.

share|cite|improve this answer

Take $N=2k+1$ then we have, $(N^2+3)/4=k^2+k+1$

$N^2/4=k^2+k+1/4,\Rightarrow $ its ceiling is $\lceil N^2/4\rceil =k^2+k+1=(N^2+3)/4$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.