a(a^n) - b(b^n) = (a-b+b)(a^n) - b(b^n)
This proof seems fairly simple, but I'm stuck. Any help would be greatly appreciated.
Let a and b be natural numbers. Show that (a^n) - (b^n) is divisible by (a-b) for all natural numbers n.
Proof. We proceed by weak mathematical induction.
Base Case. Put n=1. Then, (a^n) - (b^n) = (a^1) - (b^1) = (a - b), which is divisible by (a - b), so the base case is established.
Induction Step. Fix a natural number n and assume (a^n) - (b^n) is divisible by (a - b). Then, (a^(n+1)) - (b^(n+1)) = a(a^n) - b(b^n) = ...
I don't know what to do from here. We did similar examples in class, but we actually had numbers for a and b, so I'm not sure if I should do it like the examples or not...