1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    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...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof for x^n-y^n=(x-y)(x^n-1+ +y^n-1)
  1. X^n-y^n proof (Replies: 16)

  2. X^n<y^n n is odd (Replies: 6)

  3. Prove x^n<y^n (Replies: 1)

  4. Proof of x^n - y^n = (Replies: 6)

Loading...