Interchanging mathematical operations proof

[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 - x^2n-1y + x^2n-2y² - ... - xy^2n-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.

 

[tiny-tim]

hi homegrown898!

have you tried differentiating?

 

[homegrown898]

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

 

[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