Can the infinite sum of $\frac{2^{2k-1}}{5^{k+3}}$ be calculated with the formula of geometric series

Question

I want to calculate the infinite sum of the series below.

$$\sum_{k=2}^{\infty} \frac{2^{2k-1}}{5^{k+3}}$$

But unfortunately, I have no idea how to even start. Can I somehow use the formula of geometric series?

$$\sum_{k=2}^{\infty} ar^{k} = \frac{a}{1-r}$$

If I cannot, how should I solve the problem?

Thanks.

Answer 1

Certainly, you just got to be clever. Notice:

\begin{align*} \frac{2^{2k-1}}{5^{k+3}} &= \frac{2^{2k}}{5^k} \cdot \frac{2^{-1}}{5^3} \\ &= \frac{1}{250} \cdot \frac{4^{k}}{5^k} \end{align*}

Can you take it from here?