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For $0\leq k\leq n$ integers let $P_k(n):= n^k,\ S_k(n):= P_k(1)+\ldots P_k(n)= 1^k+\ldots n^k$. Then $P_k(0)=0$, $S_0(n)=n$. For calculate $S_1(n)$ i consider: $$P_2(n)-P_2(n-1)=2n+1$$ then $\begin{...