I'm working on the following two proofs:(adsbygoogle = window.adsbygoogle || []).push({});

1.) (x+y)^{2n+1}= x^{2n+1}+ y^{2n+1}if and only if x=0, y=0 or y=-x

and

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

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

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.

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# Interchanging mathematical operations proof

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