Let $n$ be an odd integer. Show that
$$ \left( \frac{n+1}{2} \right )^2 - \left( \frac{n-1}{2} \right )^2 = n $$
$$ \begin{align} \left( \frac{n+1}{2} \right )^2 - \left( \frac{n-1}{2} \right )^2 & = \left( \frac{n^2 + 2n +1}{4} \right ) - \left( \frac{n^2 -2n +1}{4} \right ) \\ \\ & = \left( \frac{4n}{4} \right ) \\ \\ & = n \end{align} $$