Proof for x^n-y^n=(x-y)(x^n-1+ +y^n-1)

[moviefan91]

Proof for x^n-y^n=(x-y)(x^n-1+...+y^n-1)

Homework Statement

The question asks to prove that for any n\geq1,
x^{n}-y^{n}=(x-y)(x^{n-1}+x^{n-2}y+...+y^{n-1})

Homework Equations

x^{n}-y^{n}=(x-y)(x^{n-1}+x^{n-2}y+...+y^{n-1})

The Attempt at a Solution

So far, I used induction.
So for n=1, x-y=x-y

Second step, I assume that n=k is true:
x^{k}-y^{k}=(x-y)(x^{k-1}+x^{k-2}y+...+y^{k-1})

I get stuck at n=k+1.

x^{k+1}-y^{k+1}=(x-y)(x^{k}+x^{k-1}y+...+y^{k})

When I expand RHS, I get:
x^{k+1}-x^{k}y+x^k{}y-x^{k-1}y^{2}+...+xy^{k}-y^{k+1}

I think that I need to cancel things so I can be left only with x^{k+1}-y^{k+1}, but I always have terms in the middle which do not cancel out.

What am I doing wrong?

 

[lanedance]

you need to work from the the n case to the n+1 (or vice versa)

i haven't worked it, but how about noticing:
(x+y)(x^{k} -y^{k}) = x^{k+1} -xy^{k} -x^{k}y -y^{k+1}

then you have
x^{k+1} -y^{k+1} = (x+y)(x^{k} -y^{k})-x^{k}y +xy^{k}

then see if you can work it into the required form...