The series in question is:

$\sum_{n=0}^{\infty} \frac{8 + 2^n}{5^n}$

We can break it down into two separate sums:

$\sum_{n=0}^{\infty} \frac{8}{5^n} + \sum_{n=0}^{\infty} \frac{2^n}{5^n}$

Step 1: Evaluate the first sum:

$\sum_{n=0}^{\infty} \frac{8}{5^n}$

This is a geometric series with the first term $a = 8$ and the common ratio $r = \frac{1}{5}$. The sum of an infinite geometric series is given by:

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

Substituting the values:

$S_1 = \frac{8}{1 - \frac{1}{5}} = \frac{8}{\frac{4}{5}} = 8 \times \frac{5}{4} = 10$

Step 2: Evaluate the second sum:

$\sum_{n=0}^{\infty} \frac{2^n}{5^n}$

This is also a geometric series with the first term $a = 1$ (since $\frac{2^0}{5^0} = 1$) and the common ratio $r = \frac{2}{5}$. Using the formula for the sum of an infinite geometric series:

$S_2 = \frac{1}{1 - \frac{2}{5}} = \frac{1}{\frac{3}{5}} = \frac{5}{3}$

Step 3: Combine the results:

Now, we sum both parts:

$S = S_1 + S_2 = 10 + \frac{5}{3}$

To simplify:

$S = 10 + \frac{5}{3} = \frac{30}{3} + \frac{5}{3} = \frac{35}{3}$

Thus, the value of the series is:

$\boxed{\frac{35}{3}}$

Would you like more details on any part of the process, or do you have any follow-up questions? Here are some related questions you might find interesting:

1. How does the sum formula for a geometric series apply in different contexts?
2. Can we derive similar sums for series with different exponents or more complex terms?
3. What happens if the common ratio $r$ is greater than 1?
4. How do we test the convergence of a series before summing it?
5. How can we generalize this method to handle series with multiple variables or more complex expressions?

Tip: Always check the common ratio $r$ when dealing with geometric series. If $|r| \geq 1$, the series will not converge.