Proof by Induction involving divisibility

Question

I'm trying to inductively prove:

$(a-b) | (a^n - b^n)$ where a,b are real numbers, and n is a real number and also 0

I've done my base case at zero, and made my inductive hypothesis:

$(a-b) | (a^{k+1} - b^{k-1}))$

I'm stuck here, a majority of the induction proofs I'm using involve proving something divides another, and would like some help on what to do with these. I have considered writing:

$(a^{k+1} - b^{k-1})) = s * (a-b)$, where $s$ is an element of the integers, but am still stuck there

thanks for answering!

Answer 1

Hint (if sticking to induction): $a^{k+1}-b^{k+1}=a^{k+1}-ab^k + ab^k-b^{k+1}=a(a^k-b^k)+b^k(a-b)$