Math Help - Help with divisibility proof

1. Help with divisibility proof

Let $n\geq 2$ and k be any positive integers. Prove that
$(n-1)|(n^k-1)$.

I've tried doing this by induction, but the fact that there are two variables is kind of throwing me off. Can anyone offer some advice or give me the first step or two in solving this?

In a similar problem, the book offers a hint that $n^k=((n-1)+1)^k$
Is that useful here?

2. It can be checked directly that $(n-1)(n^{k-1}+n^{k-2}+\dots+1)=n^k-1$.

3. The OP might also be interested in this generalisation:

$x, y \in \mathbb{R}$
$0 < n \in \mathbb{Z}$

$x^{n+1}-y^{n+1}=(x-y)(x^n+x^{n-1}y+\cdots+xy^{n-1}+y^n)$