How to compute $ \sum_{x=y}^\infty \binom{x}{y}((1-p)(1-q))^x$?

Question

Question

How do I compute $ \sum_{x=y}^\infty \binom{x}{y} ((1-p)(1-q))^x$? $x,y$ are integers and $p, q \in (0,1)$

Attempt

I'm trying to compute this sum to find the mixture of a geometric and binomial distribution but I can't see how I'd do it by hand.

Progress

WolframAlpha gives the result: $$-\dfrac{(1-p)^y(1-q)^y(-pq+p+q)^{-y}}{pq-p-q}$$ when $|pq-p-q+1|<1$

I would like to see how I could arrive at this answer myself.

Answer 1

Hint: Denote $u = (1 - p)(1 - q)$. Then, the sum is $u^x\sum_{k=0}^{\infty} \binom{y + k}{y} u^k$, which can be proven to be $\frac{u^x}{(1 - u)^{y + 1}}$.