Let x, y \epsilon r. use mathmatical induction to prove the identity.
x^{n+1}-y^{n+1}=(x-y)(x^{n}+x^{n-1} y++xy^{n-1}+y^{n})

Step-by-step explanation:

We will prove by mathematical induction that, for every natural n,  

We will prove our base case (when n=1) to be true:

Base case:

Inductive hypothesis:  

Given a natural n,  

Now, we will assume the inductive hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that, for y=0 the conclusion is clear. Then we will assume that

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural n,

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step-by-step explanation: