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Thread: Need help for Binomial!

  1. #1
    Nov 2012

    Need help for Binomial!

    Hi I need help with these questions on binomial theorem!

    1.) From the binomial expansion of $\displaystyle (1+x)^n$, deduce that
    $\displaystyle {2n+1 \choose 0}+{2n+1 \choose 1}+{2n+1 \choose 2}+...+{2n+1 \choose n}=4^n$

    Using T( r+1) I know the coefficient of $\displaystyle x^n$ is $\displaystyle {2n+1 \choose n}$. I had let n in $\displaystyle {n \choose r}$ be 2n+1 and x be 1.
    That way, I got $\displaystyle (2)^{2n+1}$ Is this wrong because from here, I do not know how to get $\displaystyle 4^n$

    2.) By considering the binomial expansion of $\displaystyle (1+x)^n$, find $\displaystyle \sum\limits_{r=0}^n2^r{n \choose r}$
    I got $\displaystyle 2^n$ as an answer for this one, I had just expanded out the summation and then let x=1. It's wrong..

    Thank you so much!
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  2. #2
    MHF Contributor
    Sep 2012

    Re: Need help for Binomial!

    Hey Tutu.

    For 2) consider x = 2 instead of x = 1.

    For 1) consider the binomial expansion (2 + 2)^n and how 4^n * [n]Cr gives [2n+1]Cr when you consider that adding all nCr's from 0 to n gives 2^n which means the square of this gives (2^n)^2 = 2^(2n) = (2^2)^n = 4^n.
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