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Math Help - Inductive Questions, final algebra part

  1. #1
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    Inductive Questions, final algebra part

    I need to show these are equal, but for the life of me I cant show it:
    I know they involve a little trick to show but i just cant see it

    1.
    Prove that 1/2! + 2 / 3!+ ... + n/(n+1)! = 1 - 1/(n+1)!


    (k+1)/(k+2)! = 1 - 1/(k+2)!
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by p00ndawg View Post
    I need to show these are equal, but for the life of me I cant show it:
    I know they involve a little trick to show but i just cant see it
    1.
    Prove that 1/2! + 2 / 3!+ ... + n/(n+1)! = 1 - 1/(n+1)!
    (k+1)/(k+2)! = 1 - 1/(k+2)!
    Suppose n=k is true:
    1/2! + 2 / 3!+ ... + k/(k+1)! = 1 - 1/(k+1)!

    Prove: n=k+1

    1/2! + 2 / 3!+ ... + k/(k+1)! + (k+1)/(k+2)! = 1 - 1/(k+2)!

    Left Side equals = 1 - \frac{1}{(k+1)!} + \frac{(k+1)}{(k+2)!} =

    How does this compare to 1-\frac{1}{(k+2)!}? If they are equal, which is want we want to prove, then we need to show that:

    - \frac{1}{(k+1)!} + \frac{(k+1)}{(k+2)!} = -\frac{1}{(k+2)!}

    Equivalent to:
     \frac{1}{(k+1)!} - \frac{(k+1)}{(k+2)!} = \frac{1}{(k+2)!}

    Equivalent to:
     \frac{(k+2)-(k+1)}{(k+2)!} = \frac{1}{(k+2)!}
    which is clearly true.

    I hope this helps!!
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  3. #3
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    oh man thanks!

    I didnt even THINK of subtracting the 1 at the beginning, and I thought I had tried everything!
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