How to prove $\forall n \in \mathbb{N}, x^n − y^n \text{ is divisible by } x − y$ using induction.

Question

I'm very stuck on this problem, which is an exercise from our course notes. Any clues or hints or a proof write up would be really helpful! Thanks!

Answer 1

Here's an answer with one $n$ value for the base case.

Induction step: it's true for $n=1$.

Now assume it's true for $n$; i.e., $x^{n}-y^{n}$ is divisible by $x-y$.

Since $x^{n+1}-y^{n+1}=x(x^{n}-y^{n})+y^{n}(x-y)$,

it follows that $x^{n+1}-y^{n+1}$ is divisible by $x-y$.

Answer 2

Base case: it's true for $n=0$ and $n=1$.

Induction step: assume it's true for $n-1$ and $n$;

i.e., $x^{n-1}-y^{n-1}$ and $x^n-y^n$ are divisible by $x-y$.

Then, since $x^{n+1}-y^{n+1}=(x+y)(x^n-y^n)-xy(x^{n-1}-y^{n-1})$,

it follows that $x^{n+1}-y^{n+1}$ is divisible by $x-y$.