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Summation with binomial coefficients question

  1. Apr 29, 2015 #1
    1. The problem statement, all variables and given/known data
    ##\sum\limits_{r=0}^n\frac{1}{^nC_r}=a##. Then find the value of $$\sum\sum\limits_{0\le i<j\le n}(\frac{i}{^nC_i}+\frac{j}{^nC_j})$$

    2. Relevant equations

    I have used two equations which I derived myself. This is the first one.
    20150430_000946-1-1.jpg
    The second one is:
    20150430_000435-1.jpg

    3. The attempt at a solution

    Using first equation and second equation:
    20150430_001236-1.jpg
    Now I have to subtract the cases where I=j to get the required sum. But Iis the above conclusion correct? Because I am not getting the required answer after subtracting.
     

    Attached Files:

  2. jcsd
  3. Apr 29, 2015 #2

    haruspex

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    In the last line of the derivation of your second equation, you have been inconsistent in your substitutions of m for n. Two m's should be m+1.
    But I don't understand how you use this equation anyway. a is a function of n, but where you use the equation you seem to be using it as a generic fact for any m, without changing a. I.e. you cannot now substitute n for m as being equal.
     
  4. Apr 30, 2015 #3
    Then how do you find ##(2n+1)\sum\limits_{r=0}^n\frac{r}{^nC_r}##?
     
  5. Apr 30, 2015 #4

    haruspex

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    I haven't solved it myself, yet.
    You could try concentrating on one of the two terms in the double sum. You should be able to sum that over the 'other' variable.
     
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