What is the sum of this series: $\sum_{k=n}^{\infty}(1-p)^{k-2}\cdot p$?

Question

I have: $$\sum_{k=n}^{\infty}(1-p)^{k-2}\cdot p$$ $|p|<1$.
The sum is: $$=\frac{(1-p)^{n-2}\cdot p}{1-(1-p)}=(1-p)^{n-2}$$ Or I wrong?
I use the fact that the sum of a series is: $\frac{a}{1-p}$ where $a$ is the first element and $|p|<1$.

Thank you!

Answer 1

Almost. Just be careful of the condition on $p$. For example, if $p=-0.5$ then $1-p=1.5$ and the series diverges.