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Homework Help: Proof for x^n-y^n=(x-y)(x^n-1+ +y^n-1)

  1. Sep 12, 2011 #1
    Proof for x^n-y^n=(x-y)(x^n-1+....+y^n-1)

    1. The problem statement, all variables and given/known data
    The question asks to prove that for any n[itex]\geq[/itex]1,
    [itex]x^{n}[/itex]-[itex]y^{n}[/itex]=(x-y)([itex]x^{n-1}[/itex]+[itex]x^{n-2}[/itex]y+...+[itex]y^{n-1}[/itex])


    2. Relevant equations
    [itex]x^{n}[/itex]-[itex]y^{n}[/itex]=(x-y)([itex]x^{n-1}[/itex]+[itex]x^{n-2}[/itex]y+...+[itex]y^{n-1}[/itex])


    3. 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:
    [itex]x^{k}[/itex]-[itex]y^{k}[/itex]=(x-y)([itex]x^{k-1}[/itex]+[itex]x^{k-2}[/itex]y+...+[itex]y^{k-1}[/itex])

    I get stuck at n=k+1.

    [itex]x^{k+1}[/itex]-[itex]y^{k+1}[/itex]=(x-y)([itex]x^{k}[/itex]+[itex]x^{k-1}[/itex]y+...+[itex]y^{k}[/itex])

    When I expand RHS, I get:
    [itex]x^{k+1}[/itex]-[itex]x^{k}[/itex]y+[itex]x^k{}[/itex]y-[itex]x^{k-1}[/itex][itex]y^{2}[/itex]+...+x[itex]y^{k}[/itex]-[itex]y^{k+1}[/itex]

    I think that I need to cancel things so I can be left only with [itex]x^{k+1}[/itex]-[itex]y^{k+1}[/itex], but I always have terms in the middle which do not cancel out.

    What am I doing wrong?
     
  2. jcsd
  3. Sep 12, 2011 #2

    lanedance

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    Homework Helper

    Re: Proof for x^n-y^n=(x-y)(x^n-1+....+y^n-1)

    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:
    [tex](x+y)(x^{k} -y^{k}) = x^{k+1} -xy^{k} -x^{k}y -y^{k+1}[/tex]

    then you have
    [tex]x^{k+1} -y^{k+1} = (x+y)(x^{k} -y^{k})-x^{k}y +xy^{k}[/tex]

    then see if you can work it into the required form...
     
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