The sum you're asking for is an infinite series:

$\sum_{n=1}^{\infty} \frac{2^n + 5^n}{8^n}$

We can break this into two separate geometric series:

$\sum_{n=1}^{\infty} \frac{2^n}{8^n} + \sum_{n=1}^{\infty} \frac{5^n}{8^n}$

First Series:

$\sum_{n=1}^{\infty} \left(\frac{2}{8}\right)^n = \sum_{n=1}^{\infty} \left(\frac{1}{4}\right)^n$ This is a geometric series with the first term $a = \frac{1}{4}$ and the common ratio $r = \frac{1}{4}$.

For a geometric series $\sum_{n=1}^{\infty} ar^n$, the sum is:

$S = \frac{r}{1 - r}$

Thus, the sum of the first series is:

$S_1 = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}$

Second Series:

$\sum_{n=1}^{\infty} \left(\frac{5}{8}\right)^n$ This is also a geometric series with the first term $a = \frac{5}{8}$ and the common ratio $r = \frac{5}{8}$.

The sum of this series is:

$S_2 = \frac{\frac{5}{8}}{1 - \frac{5}{8}} = \frac{\frac{5}{8}}{\frac{3}{8}} = \frac{5}{3}$

Total Sum:

Now, adding the two series together:

$S = S_1 + S_2 = \frac{1}{3} + \frac{5}{3} = \frac{6}{3} = 2$

Thus, the sum of the infinite series is:

$\boxed{2}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you derive the sum of an infinite geometric series?
2. Can an infinite geometric series converge if the common ratio is greater than 1?
3. What is the general form for the sum of geometric series starting from $n = 1$?
4. How does the value of the common ratio affect the speed of convergence of a geometric series?
5. Can a geometric series represent real-world phenomena?

Tip: Geometric series converge only if the absolute value of the common ratio is less than 1.