Which of the following relations are true?Here $[x]$ denotes the greatest integer less then or equal to $x$ (as in option b) symbol

Question

Which of the following relations are true?

$1)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{\frac{n(n+1)}{2}}$

$2)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{[\frac{n}{2}]}$

$3)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{n^2}$

Here $[x]$ denotes the greatest integer less then or equal to $x$ (as in option b) symbol

My attempts : if I take $n=0$ then all option $1,2,3$ are True

Any other logics?

Answer 1

If $n=4k+1$ then last one is not true.

Also $$(-1)^{{n(n+1)\over 2}} = (-1)^{{n(n-1)\over 2}+n}=(-1)^{{n(n-1)\over 2}}(-1)^n$$

so for $n$ is odd it is not true.

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So, only the second can be always true:

If $n=2k$ then $$(-1)^{k(2k-1)} = (-1)^k = (-1)^{[\frac{n}{2}]}$$ If $n=2k+1$ then $$(-1)^{(2k+1)k} = (-1)^k = (-1)^{[\frac{n}{2}]}$$