Calculate the sum $\sum_{k=0}^{p}\binom{p-1+k}{p-1}x^{k}$

Question

Let $p>5$ a natural number and let $x$ be a rational number, $0<x<1$. Would anyone know how to calculate the following sum: $$\sum_{k=0}^{p}\binom{p-1+k}{p-1}x^{k}?$$

Answer 1

Hint :

Coefficient of $t^{p-1}$ in $$\left( (1+t)^{p-1}x^0+ (1+t)^{p}x^1+(1+t)^{p+1}x^2\ldots +(1+t)^{2p-1}x^p\right) $$

Which is a finite GP which can be summed easily.