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Math Help - Need help with a proof using factorials

  1. #1
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    Need help with a proof using factorials

    Hey,

    Could someone please help me out here:

    I need to prove that

    (2n)! / n!(n+1)! is a natural number for every n∈N.


    I tried already with induction, but got stuck.

    Thanks.
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  2. #2
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    Quote Originally Posted by zidus76 View Post
    Hey,

    Could someone please help me out here:

    I need to prove that

    (2n)! / n!(n+1)! is a natural number for every n∈N.


    I tried already with induction, but got stuck.

    Thanks.

    you need induction for this... where did you get stuck?
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  3. #3
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    On the 3rd step, I plug in (n+1) and get

    (2n+1)!/(n+1)!(n+2)!

    Then I transform it to

    (2n+1)2n!/n!(n+1)!(n+1)(n+2)

    And since I assume that (2n)!/n!(n+1)! is a natural number I make it =k, then I got

    k* (2n+1)/(n+1)(n+2)

    If I could prove that (2n+1)/(n+1)(n+2) is also a natural number I would be done, but I can't do anything with (2n+1)/(n+1)(n+2).
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  4. #4
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    yes sorry what you said cannot be proved via induction, you corrected your mistake

    (i should read questions fully sorry)
    Last edited by artvandalay11; October 11th 2009 at 04:54 PM. Reason: incorrect
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  5. #5
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    Can't, since if you plug in 1 in the first step you get 1/2 and that's not a natural number. Anyway, I realized I made a computing mistake, it should be (2n+2)!, not (2n+1)!. Then I get (2n+1)(2n+2)/(n+1)(n+2), then I can cancel out (n+1) and then I get 2*(2n+1)/(n+2). But I still can't do anything with that, tried induction as well.

    I also just read that these numbers are also known as the Catalan numbers, if it helps anyone. I couldn't find any proofs on the internet.
    Last edited by zidus76; October 11th 2009 at 06:22 PM.
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