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Math Help - Induction question, not sure how to do this, help appreciated

  1. #1
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    Induction question, not sure how to do this, help appreciated

    Use induction to show that:

    (1+x) E (n, i=0) (-x)^i = 1 + x^(n+1)

    The E (n, i=0) bit is the summation notation with the n on top and i=0 on the bottom, you know?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by brumby_3 View Post
    Use induction to show that:

    (1+x) E (n, i=0) (-x)^i = 1 + x^(n+1)

    The E (n, i=0) bit is the summation notation with the n on top and i=0 on the bottom, you know?
    I assume you meant you want to show  \displaystyle (x+1)\sum_{i=0}^n (-x)^i=(-1)^nx^{n+1}+1

    I leave the base case to you.

    Suppose your assertion is true for  n i.e.  \displaystyle (x+1)\sum_{i=0}^n (-x)^i=(-1)^nx^{n+1}+1

    Now  \displaystyle (x+1)\sum_{i=0}^{n+1} (-x)^i=(x+1)(-x)^{n+1}+(x+1)\sum_{i=0}^{n+1} (-x)^i=(x+1)(-x)^{n+1}+(-1)^nx^{n+1}+1

    Can you finish up from here?
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