for all positive natural numbers n and for any integers a and b such that a != b a^n - b^n is divisible by a-b

the induction hypothesis is to assume the a^k - b^k is divisible by a-b

I need to show that a^(k+1) - b^(k+1) is divisible by (a-b). I have seen the solution floating around on the internet. The expression can be manipulated into the form

a(a^k - b^k) + b^k(a -b) which is divisible by (a-b) both directly in the second term and by the induction hypothesis in the first. I don't necessarily want to know how to get to this result. But how do I THINK about how to get to the result?

Any advice would be helpful. Thanks