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Math Help - binomial coefficients

  1. #1
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    binomial coefficients

    if  C_r =\frac {n!}{(n-r)!(r)!}

    prove

     \frac{C_1}{C_0} + 2\frac{C_2}{C_1} +3\frac{C_3}{C_2} + ... +n\frac{C_n}{C_(n-1)} = \frac{n(n+1)}{2}
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  2. #2
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    use \frac{C_{r}}{C_{r-1}} = \frac{n[n-(r-1)](r-1)!}{r!}
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  3. #3
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    Also try to link in the fact that

    1+2+3+\cdots+(n-1)+n=S [1]

    where S is the sum of terms, then making the order of the LHS opposite

    n+(n-1)+\cdots+3+2+1 = S [2]

    adding [1]+[2] term for term gives

     (n+1)+(n+1)+(n+1)+\cdots+(n+1)+(n+1) = 2S

     n(n+1) = 2S

     \frac{n(n+1)}{2} = S
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