$a-b$ divides $a^n-b^n$

You can use $$a^{k+1}-b^{k+1}=a(a^{k}-b^{k})+b^{k}(a-b).$$

Add and subtract $ab^n$ from $a^{n+1} - b^{n+1}$.

Note that, $aa^{n} - ab^n + ab^n - bb^{n}$ = $a(a^n - b^n) + (a-b)b^n$. Now you can easily see that this term is divisible by $(a-b)$.

Hint:

Try to use Euclid's division algorithm to divide the polynomial $$X^k-1$$ by $X-1$. Do you see a pattern for different $k$'s?

If so, use the induction to prove your hypothesis. You get something of the form $$X^k-1 = (X-1)Q(X)$$ for some polynomial $Q(X)$. Let now $X = \frac{a}{b}, b\neq 0$.