1. ## please prove these binomial identities?

I have been searching the web for the past three days but can't find the algebraic or inductive proof anywhere. Please help me, i have no idea on how to prove these.

Note: if you are not gonna solve it, then don't comment.

THANKS

2. ## Re: please prove these binomial identities?

Originally Posted by szak1592
Note: if you are not gonna solve it, then don't comment.

THANKS
The purpose for this forum is to provide help so that people can learn the material on their own. I know this is not the response you wanted. But, I am willing to help you solve these problems. I just wanted to let you know that statement is not really appropriate for this forum.

What is Formula (8)? I assume it is the definition of the general binomial coefficient. By definition:

\begin{align*}\binom{-m}{k} & = \dfrac{-m(-m-1)\cdots (-m-k+1)}{k!} \\ & = (-1)^k \dfrac{(m+k-1)(m+k-2)\cdots (m+1)m}{k!} \\ & = (-1)^k \dfrac{(m+k-1)(m+k-2)\cdots (m+1)m}{k!} \cdot \dfrac{(m-1)!}{(m-1)!} \\ & = (-1)^k \dfrac{(m+k-1)!}{k!(m-1)!} \\ & = (-1)^k \binom{m+k-1}{k}\end{align*}

For the second equation, we can use Pascal's Rule: $\binom{n+1}{k} = \binom{n}{k} + \binom{n}{k-1}$.

Proof by induction: $\sum_{s=0}^{1-1} \binom{k+s}{k} = \binom{k+0}{k} = \binom{k}{k} = 1 = \binom{k+1}{k+1}$, so the formula holds when $k \ge 0, n = 1$. Assume the formula holds for all positive integers up to $n$.

\begin{align*}\sum_{s=0}^{(n+1)-1} \binom{k+s}{k} & = \sum_{s = 0}^{n-1} \binom{k+s}{k} + \binom{k+(n+1)-1}{k} \\ & = \binom{k+n}{k+1} + \binom{k+n}{k} \qquad \text{(by the induction assumption)} \\ & = \binom{k+n+1}{k+1} \qquad \text{(by Pascal's Rule)}\end{align*}

So, by the principle of mathematical induction, the formula holds for all positive integers $n$.

3. ## Re: please prove these binomial identities?

thanks a lot.....I know that forums are to help people not do their homework for them.....but it's just that I have been looking everywhere....youtube lectures, about a dozen pdfs and ppts but I just could not find these two proofs, there were three others but I found them. Anyways thank you so much!!!