Prove this binomial identity using induction

Question

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?

Answer 1

You are in the right direction,

On solving for $$\sum_{n\ge 0}\binom{n+k}kx^n\;.$$

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

And that is what you needed to prove. You are done:)

Answer 2

You’re practically done: just solve your equation for

$$\sum_{n\ge 0}\binom{n+k}kx^n\;.$$