Find the closed form solution to a geometric series not starting at 0

Question

I have the following geometric series:
$$
\sum_{i=k_0+1}^{k}{(b^\epsilon)^i}
$$

How can I find a closed-form solution for the following formula?
I know it's a geometric series, but in order to apply the formula, I need it to start from 0 and the $k_0+1$ is confusing me

Thanks

Answer 1

Let's use the following notation: $$ \sum_{i=k_0+1}^{k}{(b^\epsilon)^i} = \sum_{i=k_0+1}^{k}{C^i} =S $$ Therefore we can say that: $$ S = C^{k_0+1} + C^{k_0+2} + ... + C^{k}\\ C*S = C^{k_0+2} + C^{k_0+3} + ... + C^{k+1}\\ C*S - S = S(C-1) = C^{k+1} - C^{k_0+1}\\ S = \frac{C^{k+1} - C^{k_0+1}}{C-1} $$