# Interchanging mathematical operations proof

1. Jan 31, 2012

### homegrown898

I'm working on the following two proofs:

1.) (x+y)2n+1 = x2n+1 + y2n+1 if and only if x=0, y=0 or y=-x

and

2.) (x+y)2n = x2n + y2n if and only if x=0 or y=0

I've tried using induction and get stuck at a certain point. I've also tried playing around with summations since there is a binomial expansion in there. I haven't had any luck. Earlier today however I found that this equality is true:

x2n+1 + y2n+1 = (x+y)(x2n - x2n-1y + x2n-2y2 - ... - xy2n-1 + y2n)

I haven't played around with this yes, but I think setting this equal to (1) might help prove this.

I'm just wondering if anyone has any ideas on this.

Last edited: Jan 31, 2012
2. Jan 31, 2012

### tiny-tim

hi homegrown898!

have you tried differentiating?

3. Jan 31, 2012

### homegrown898

I haven't tried that no and I'm not exactly sure what you mean

4. Jan 31, 2012

### tiny-tim

if eg (1 + x)6 - 1 - x6 = 0 at x = 0 and at x = something else,

then its derivative must be zero somewhere in between