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Math Help - problem on binomial

  1. #1
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    problem on binomial

    Please solve the problem of Binomial theory.
    1) Prove that C0+4C1+8C2+12C3+..+4nCn=1+n2n-1
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  2. #2
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    Quote Originally Posted by somnath6088 View Post
    Please solve the problem of Binomial theory.
    1) Prove that C0+4C1+8C2+12C3+..+4nCn=1+n2n-1
    Are those C's constants or sequences?
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  3. #3
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    problem on binomial

    Please solve the problem of Binomial theory.
    Prove that C0C1+C1C2+C2C3++Cn-1Cn= (2n)!/(n+1)!(n-1)!
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    The C0 implies C(subscript 0).Which are binomial constants.
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  5. #5
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    Hello,

    Please use (n,p) if you mean pCn, like you wrote...

    And the latter term is not understandable...
    use parenthesis, power ^, etc...
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  6. #6
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    Quote Originally Posted by somnath6088 View Post
    prove that

    nC0 + 4.nC1 + 8.nC2 + 12.nC3 +...........+4n.nCn=1+n2^(n+1)
    Since  {n \choose 0} = 1, you only need to prove the following:

    4 \, {n \choose 1} + 8 \, {n \choose 2}  + 12 \, {n \choose 3} + ..... + 4n \, {n \choose n} = n 2^{n+1}

    Note that:

    (1 + x)^n = \sum_{k=0}^{n} {n \choose k} \, x^k .... (1)

    Differentiate both sides of (1) with respect to x:

    n (1 + x)^{n-1} = \sum_{k=0}^{n} {n \choose k} \, k\, x^k .... (2)

    (Identity (2) can also be proved using a combinatorial argument)

    Now substitute x = 1 into (2). Then multiply both sides by 4 and you get the result.
    Last edited by mr fantastic; June 18th 2008 at 03:07 AM. Reason: Added that identity (2) could also be proved using a combinatorial argument
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  7. #7
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    Quote Originally Posted by somnath6088 View Post
    Please solve the problem of Binomial theory.
    1) Prove that C0+4C1+8C2+12C3+……..+4nCn=1+n2n-1
    Is it meant to read

    {n \choose 0} + 4 \, {n \choose 1} + 8 \, {n \choose 2} + 12 \, {n \choose 3} + ..... + 4n \, {n \choose n} = 1 + n 2^{n-1} ?

    This statement is FALSE. Substitute n = 1, for example, to see this.

    The correct identity is given (and a solution suggested) in this thread: http://www.mathhelpforum.com/math-he...l-problem.html
    Last edited by mr fantastic; June 18th 2008 at 03:38 AM. Reason: Added the hyperlink
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  8. #8
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    Quote Originally Posted by somnath6088 View Post
    Please solve the problem of Binomial theory.
    Prove that C0C1+C1C2+C2C3+……………+Cn-1Cn= (2n)!/(n+1)!(n-1)!
    Translation:

    Prove that {n \choose 0} \, {n \choose 1} + {n \choose 1} \, {n \choose 2} + {n \choose 2} \, {n \choose 3} + \, ..... \, + {n \choose n-1} \, {n \choose n} = \frac{(2n)!}{(n+1)! \, (n - 1)!}.


    Three hints:

    #1. Note that the right hand side is equivalent to {2n \choose n-1}. This suggests hint #2.

    #2. Consider Vandermonde's Identity.

    #3. Note that {n \choose k} = {n \choose n - k}.
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