Compute $\sum_{n=0}^{\infty}{\frac{2^n+5^n}{7^{n-2}}}$

Question

i tried further simplifying the sum, and this is what I came up with:$$\sum_{n=0}^{\infty}{\frac{2^n+5^n}{7^{n-2}}}=\sum_{n=0}^{\infty}{\left(\frac{5}{7}\right)^n\cdot 49\left(1+\left(\frac{2}{5}\right)^n\right)}$$ How should I continue? Usually I am given a rational expression where I can perform partial fraction decomposition.

Answer 1

It is equal to$$7^2\left(\sum_{n=0}^\infty\left(\frac27\right)^n+\sum_{n=0}^\infty\left(\frac57\right)^n\right).$$Can you take it from here?