Let $\displaystyle n(k)$ be defined as the highest power dividing $\displaystyle (p^k)!$. That is $\displaystyle (p^ {n(k)} \mid (p^k)!$ but $\displaystyle p^{n(k)+1} \nmid (p^k)!]$. Prove that $\displaystyle n(k)=1+p+p^{2}+....+p^{k}$