Let's analyze the geometric series given in the image:

The series is: $(x + 6) + (x + 12) + (x + 15) + \dots$

This is a geometric series, so we need to identify:

1. The First Term ($a$): The first term of the series is $x + 6$.

2. The Common Ratio ($r$): To find the ratio, we divide the second term by the first term: $r = \frac{x + 12}{x + 6}$

3. The Fifth Term: We can use the formula for the $n$-th term of a geometric sequence, $a_n = a \cdot r^{n-1}$, to find the fifth term.

4. Sum of the First Five Terms: The sum of the first $n$ terms in a geometric sequence is given by: $S_n = a \frac{1 - r^n}{1 - r} \quad \text{(for } r \neq 1\text{)}$ Substitute $n = 5$ here to get the sum of the first five terms.

5. The Infinite Sum: If $|r| < 1$, we can calculate the infinite sum using: $S_\infty = \frac{a}{1 - r}$

Would you like me to continue with the calculations assuming specific values, or is there a particular part you would like clarified?

Here are five related questions for further exploration:

1. What is the value of $x$ if we know the common ratio?
2. How does changing $x$ affect the sum of the series?
3. Can we determine the convergence of this series for different values of $x$?
4. How would the series change if each term had an additional constant added?
5. What if the sequence was arithmetic instead of geometric?

Tip: Remember, for a geometric series to converge (have an infinite sum), the common ratio $r$ must be between -1 and 1.