Solving Series of a Sum.

Question

Hey Guys I was wondering if I could get some help in solving a series. $$\sum_{n={0}}^\infty\frac{(n-1)!}{(k-1)!(n-k)!}(\frac{1}{2})^{n+1}$$ How would I go on about and solve this series? If I could get an explanation I would greatly appreciate it and one more thing the use for this function is used for the calculation for a solera method where $n$ is the amount of years the solera has been going and $k$ is the number of tiers where in the example I am doing there is only 5 tiers, so it will always be $k=5$.

Answer 1

Hint: for $|x| \lt 1$,

$$\sum_{n=k}^{\infty} \binom{n-1}{k-1} x^{n-k} =\sum_{n=0}^{\infty} \binom{n+k-1}{k-1} x^n = (1-x)^{-k}$$