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Interchanging mathematical operations proof

  1. Jan 31, 2012 #1
    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. jcsd
  3. Jan 31, 2012 #2

    tiny-tim

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    hi homegrown898! :smile:

    have you tried differentiating?
     
  4. Jan 31, 2012 #3
    I haven't tried that no and I'm not exactly sure what you mean
     
  5. Jan 31, 2012 #4

    tiny-tim

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    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 :wink:
     
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