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  1. #1
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    Show that...

    Show that \frac{1}{r!}-\frac{1}{(r+1)!} = \frac{r}{(r+1)!}

    thanks!
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  2. #2
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    Quote Originally Posted by BabyMilo View Post
    Show that \frac{1}{r!}-\frac{1}{(r+1)!} = \frac{r}{(r+1)!}

    thanks!
    (r+1)! = (r+1) * r!

    So lcm(r!, (r+1)!) divides (r+1)!

    We can add the fractions on the LHS using the denominator (r+1)!

    You should quickly see why it works out.

    By the way, LHS means left hand side of the equation, and if you didn't understand the line about lcm, it's not terribly important, as long as you can see why (r+1)! can be used as a common denominator.

    Edit:

    Actually, the stronger statement

    lcm(r!, (r+1)!) = (r+1)!

    holds. I just wrote the weaker statement because I knew it was safe, without having to think. But obviously lcm(a, b) >= max(a, b) so it was a pretty silly precaution.
    Last edited by undefined; April 21st 2010 at 06:28 AM.
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  3. #3
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    Quote Originally Posted by undefined View Post
    (r+1)! = (r+1) * r!

    So lcm(r!, (r+1)!) divides (r+1)!

    We can add the fractions on the LHS using the denominator (r+1)!

    You should quickly see why it works out.

    By the way, LHS means left hand side of the equation, and if you didn't understand the line about lcm, it's not terribly important, as long as you can see why (r+1)! can be used as a common denominator.
    thank you but how do i know (r+1)! = (r+1) * r! ?
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  4. #4
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by BabyMilo View Post
    thank you but how do i know (r+1)! = (r+1) * r! ?
    Consider

    5! = 5 * 4 * 3 * 2 * 1

    6! = 6 * 5 * 4 * 3 * 2 * 1

    It should be apparent that 6! = 6 * 5!
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  5. #5
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    Quote Originally Posted by BabyMilo View Post
    thank you but how do i know (r+1)! = (r+1) * r! ?
    Because that follows directly from the definition of n!.
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  6. #6
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    Because that follows directly from the definition of n!.
    Yes there are multiple definitions of n!, one of them is recursive. I didn't bring it up in case the OP was only familiar with the non-recursive definition.
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