How to 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?

Answer 2

Just remember that $$\displaystyle\sum_{n\ge0} (r^n + s^n) = \displaystyle\sum_{n\ge0} r^n + \displaystyle\sum_{n\ge0} s^n$$ because $|r|,|s|<1$ and then both $r^n$ and $s^n$ converges.

Answer 3

And recall that $$\sum_{n=0}^\infty r^n=\frac{1}{1-r}$$ whenever $-1<r<1$.