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Math Help - binomial theorem questions.

  1. #1
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    binomial theorem questions.

    Can anyone please show me the the proof of the ff?


    The summation of (nCr)2^r <as r goes from 0 to n> = [3^n + (-1)^n]/2



    The summation of r(nCr) <as r goes from 1 to n>= n(2^n-1)
    (combinatorial proof)


    Your help would be very much appreciated. Thanks.
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  2. #2
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    Don't you mean

    [LaTeX ERROR: Convert failed]

    or


    [LaTeX ERROR: Convert failed]

    These two results follow directly from the application of the binomial theorem in reverse.

    [LaTeX ERROR: Convert failed]

    with x = 1 and y = 2 where in the second one we are subtracting 1 since we are not summing over the first term which is [LaTeX ERROR: Convert failed]

    EDIT: Here is the second one:

    [LaTeX ERROR: Convert failed]

    [LaTeX ERROR: Convert failed]


    Now the sum becomes:


    [LaTeX ERROR: Convert failed]
    Last edited by Vlasev; August 10th 2010 at 08:11 PM.
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