Results 1 to 3 of 3
Like Tree4Thanks
  • 3 Post By johng
  • 1 Post By Idea

Thread: Proving combinatoric identity

  1. #1
    Newbie
    Joined
    Mar 2015
    From
    USA
    Posts
    16

    Proving combinatoric identity

    Hi,

    I need to prove the following identity:
    Proving combinatoric identity-q15.jpg

    I thought it's somewhat similar to the Newton's binomial theorem, but I coudn't find how to solve this using this theorem.
    Do you have an idea how can I solve this in any algebric/combinatoric way?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2012
    From
    Athens, OH, USA
    Posts
    1,074
    Thanks
    444

    Re: Proving combinatoric identity

    If you know the calculus, there's a simple proof of your formula. However, unfortunately, I couldn't find any purely algebraic proof.

    For n any non-negative integer and x any real,
    $$(1+x)^n=\sum_{i=0}^nC_i^nx^i$$
    Now integrate each side of the above equation from 0 to 1; first the integral of the left side:
    $$\int_0^1(1+x)^n\,dx={(1+1)^{n+1}\over n+1}-{(1+0)^{n+1}\over n+1}={2^{n+1}-1\over n+1}$$
    Now the integral of the right side:
    $$\sum_{i=0}^nC^n_i\int_0^1x^i\,dx=\sum_{i=0}^nC^n _i{1^{i+1}-0^{i+1}\over i+1}=\sum_{i=0}^n{C^n_i\over i+1}$$
    Thanks from topsquark, Plato and romsek
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    643
    Thanks
    292

    Re: Proving combinatoric identity

    1+\sum _{k=0}^n \frac{n+1}{k+1}C_k^n=1+\sum _{k=0}^n C_{k+1}^{n+1}=

    1+\sum _{k=1}^{n+1} C_k^{n+1}=\sum _{k=0}^{n+1} C_k^{n+1}=2^{n+1}
    Thanks from johng
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with proving an identity.
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: Feb 13th 2011, 02:19 PM
  2. Proving an identity that's proving to be complex
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Jul 21st 2009, 01:30 PM
  3. Proving an identity
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: Jul 2nd 2009, 09:08 PM
  4. Need help with proving an identity!
    Posted in the Trigonometry Forum
    Replies: 13
    Last Post: Apr 24th 2008, 11:12 PM
  5. combinatoric identity
    Posted in the Statistics Forum
    Replies: 1
    Last Post: Feb 17th 2007, 05:24 PM

/mathhelpforum @mathhelpforum