Results 1 to 3 of 3

Thread: Binomial Formula

  1. #1
    Junior Member
    Joined
    Aug 2007
    Posts
    30

    Binomial Formula

    Prove, that for every integer n>1,

    $\displaystyle \binom{n}{1}$ - 2$\displaystyle \binom{n}{2}$ +3$\displaystyle \binom{n}{3}$ +...+$\displaystyle (-1)^{n-1}$n$\displaystyle \binom{n}{n}$=0

    I get $\displaystyle \sum_{x=0}^{n}$$\displaystyle (-1)^{x-1}$x$\displaystyle \binom{n}{x}$

    The teacher gave us a hint that said we should differentiate and then plug in x =-1. I'm having a hard time converting this to the binomial theorem and understand why x=-1 shows it for all 'n'
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Oct 2007
    From
    London / Cambridge
    Posts
    591
    $\displaystyle (1+x)^n = 1 + \binom{n}{1} x + \binom{n}{2} x^2 + \binom{n}{3} x^3 + \binom{n}{4} x^4 ..... \binom{n}{n} x^n$


    differentiate both sides with respect to x

    $\displaystyle n(1+x)^{n-1} = \binom{n}{1} + 2 \binom{n}{2} x +3 \binom{n}{3} x^2 + 4 \binom{n}{4} x^3 ..... n \binom{n}{n} x^{n-1}$

    sub x = -1

    $\displaystyle n(1-1)^{n-1} = \binom{n}{1} - 2 \binom{n}{2} +3 \binom{n}{3} - 4 \binom{n}{4} ..... + (-1)^{n-1} n \binom{n}{n}$


    $\displaystyle 0 = \binom{n}{1} - 2 \binom{n}{2} +3 \binom{n}{3} - 4 \binom{n}{4} ..... + (-1)^{n-1} n \binom{n}{n}$
    Last edited by bobak; Feb 20th 2008 at 01:41 AM. Reason: fixed typo, Thank you James Bond
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2007
    Posts
    329
    Just a little typo: $\displaystyle (-1)^{n-1}$ twice...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. binomial formula, probability problem.
    Posted in the Statistics Forum
    Replies: 1
    Last Post: Oct 12th 2011, 12:56 AM
  2. Trouble with a binomial distribution formula
    Posted in the Statistics Forum
    Replies: 4
    Last Post: Mar 28th 2011, 05:46 PM
  3. Binomial Formula, solving n?
    Posted in the Statistics Forum
    Replies: 0
    Last Post: Feb 16th 2010, 10:32 AM
  4. Binomial Coefficients Formula
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Dec 28th 2009, 04:32 PM
  5. BiNomial Distribution Formula
    Posted in the Statistics Forum
    Replies: 1
    Last Post: Dec 1st 2008, 10:51 PM

Search Tags


/mathhelpforum @mathhelpforum