Summation with binomial coefficients question

[AdityaDev]

Homework Statement

$\sum\limits_{r=0}^n\frac{1}{^nC_r}=a$. Then find the value of $$\sum\sum\limits_{0\le i<j\le n}(\frac{i}{^nC_i}+\frac{j}{^nC_j})$$

Homework Equations

I have used two equations which I derived myself. This is the first one.

The second one is:

3. The Attempt at a Solution
Using first equation and second equation:

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.

 

[haruspex]

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.

 

[AdityaDev]

Then how do you find $(2n+1)\sum\limits_{r=0}^n\frac{r}{^nC_r}$?

 

[haruspex]

Then how do you find $(2n+1)\sum\limits_{r=0}^n\frac{r}{^nC_r}$?

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.