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Divisibility of Sequences

Let m, n be in N, with m, n >= 1 and n odd.

Let S_m = 1^n + 2^n + 3^n + ... + m^n.

Prove that S_m is divisible by 1+2+...+m.

Solution Preview

As we know that 1+2+...=m=m(m+1)/2, and gcd(m,m+1)=1, then we have two cases:
case 1: m is odd, then m+1 is even, we need to show that both m and (m+1)/2 divides S_m
case 2: m is even, then m+1 is odd, we need to show that both m/2 and m+1 divides S_m
We have a basic fact that a+b divides a^n+b^n for ...

Solution Summary

The divisibility of a sequence is investigated.