1. if $\displaystyle a^2+b^2 = c^2 $ ,please show why $\displaystyle (a+b) $ and $\displaystyle c $ have the same parity?
2. proof this using a combinatorial argument:
$\displaystyle \sum\limits_{r=0}^{n}C_r^n 2^{r} = 3^{n} $
1)$\displaystyle (a+ b)^2= a^2+ 2ab+ b^2$
"2ab" is clearly even so the parity of $\displaystyle (a+ b)^2$ is the same as the partity of $\displaystyle c^2= a^2+ b^2$.
And, of course, since $\displaystyle (2n)^2= 4n^2$ and $\displaystyle (2n+ 1)^2= 4n^2+ 4n+ 1= 2(2n^2+ 2n)+ 1$ the parity of $\displaystyle a^2$ is the same as the parity of a.