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Math Help - factorial proof - please help

  1. #1
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    factorial proof - please help

    Thanks for taking my question:

    How can I prove by induction the following:

    summation E (i=1 to n) of i * i! = (n+1)! -1

    - thanks
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  2. #2
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    Quote Originally Posted by Rayk2003 View Post
    Thanks for taking my question:

    How can I prove by induction the following:

    summation E (i=1 to n) of i * i! = (n+1)! -1

    - thanks
    Hi,
    is it \sum_{i=1}^n i\cdot i! = (n+1)!-1 ?

    base case: n=1 ... 1\cdot 1!=(1+1)!-1 verified.

    inductive step: suppose \sum_{i=1}^n i\cdot i! = (n+1)!-1 and let's prove it holds for n+1 too, i.e. we want to prove \sum_{i=1}^{n+1} i\cdot i!=((n+1)+1)!-1.

    \sum_{i=1}^{n+1} i\cdot i! = \sum_{i=1}^n i\cdot i! + (n+1)\cdot (n+1)!=(n+1)!-1+(n+1)\cdot (n+1)! =
    =(n+1)!(1+(n+1))-1=(n+1)!(n+2)-1=(n+2)!-1=((n+1)+1)!-1 inductive step verified.
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