Proving that for any odd integer: $\lceil \frac{N^2}{4} \rceil = \frac{N^2 + 3}{4}$

Question

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!

Answer 1

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

Answer 2

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

Answer 3

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 $\lceil N^2/4\rceil =k^2+k+1=(N^2+3)/4$