1. ## Binomial Expansion proof

I have a proof that I cannot complete. Have done as far as I can go and am now stuck. The RHS is fine, just no idea about the LHS.
Just not sure how to proceed now.

help very much appreciated.
jacs

2. ## Binomial Proof

Hello jacs
Originally Posted by jacs
I have a proof that I cannot complete. Have done as far as I can go and am now stuck. The RHS is fine, just no idea about the LHS.
Just not sure how to proceed now.

help very much appreciated.
jacs
You are OK so far. Now plug the value of $\displaystyle c$ back in, and put $\displaystyle x =1$ as in the first proof:

$\displaystyle \frac{2^{n+1}}{n+1}-\frac{1}{n+1}=^nC_0+\tfrac12^nC_1+...+\tfrac{1}{n+ 1}^nC_n =\sum_{r=0}^n\frac{^nC_r}{r+1}$

$\displaystyle \Rightarrow \frac{2^{n+1}-1}{n+1}=\sum_{r=0}^n\frac{^nC_r}{r+1}$

Now note that if we replace n by $\displaystyle (n+1)$ in the first result, $\displaystyle \sum_{r=1}^n{^nC_r}=2^n-1$, we get $\displaystyle \sum_{r=1}^{n+1}{^{n+1}C_r}=2^{n+1}-1$, and the result follows.

$\displaystyle ^{n+1}C_a=\frac{1}{a}(n+1)^nC_{a-1}$