# Prove this binomial identity using induction

prove this identity:

$(1-x)^{-k} = \sum\limits_{i>=0} \binom {n+k-1} {k-1} x^n$

using induction.

Verification for k=1 is trivial. assuming k= i, proving the identity when k=i+1 is something i am having problem with.

This is what i have done:

\begin{align*} \sum_{n\ge 0}\binom{n+k}kx^n&=\sum_{n\ge 0}\left(\binom{n+k-1}{k-1}+\binom{n+k-1}k\right)x^n\\ &=(1-x)^{-k}+\sum_{n\ge 0}\binom{n+k-1}kx^n\\ &=(1-x)^{-k}+\sum_{n\ge 1}\binom{n+k-1}kx^n\\ &=(1-x)^{-k}+x\sum_{n\ge 1}\binom{n+k-1}kx^{n-1}\\ &=(1-x)^{-k}+x\sum_{n\ge 0}\binom{n+k}kx^n\;. \end{align*}

how do i finish the proof?

On solving for $$\sum_{n\ge 0}\binom{n+k}kx^n\;.$$
You get that $$\sum_{n\ge 0}\binom{n+k}kx^n\;= (1-x)^{-k-1}$$
$$\sum_{n\ge 0}\binom{n+k}kx^n\;.$$