Fix an integer $n\in \Bbb{Z}^+$. For non-negative integer $r$ define the sum $$S_r = 1^r+2^r+...+n^r.$$
From the equality $$(m+1)^{k+1}-m^{k+1} = \binom{k+1}{1}m^k+...+\binom{k+1}{k}m+1$$ sum up both LHS and RHS for $m=1,2,...,n$ gives the following:
$$(n+1)^{k+1}-1 = \binom{k+1}{1}S_{k}+\binom{k+1}{2}S_{k-1}...+\binom{k+1}{k}S_1+S_0.$$
This is a recursive definition for $S_k$, i.e., if I wish to find the sum $S_k=1^k+...+n^k$ I could have find $S_{k-1},...,S_1,S_0$. But how could one possibly find the explicit expression of $S_k$ from the recursive one? I have seen generating function but I really don't get how one could apply it on this problem.
