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$\geq$1,
$x^{n}$-$y^{n}$=(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}$+...+x$y^{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...