The image asks us to evaluate the sum of a geometric series:

$S = \sum_{n=1}^{5} 6(2)^{n-1}$

This is a geometric series with:

• First term $a = 6$
• Common ratio $r = 2$
• Number of terms $n = 5$

We can use the formula for the sum of a geometric series:

$S = \frac{a(1 - r^n)}{1 - r}$

Step-by-step solution:

1. First term $a$: The first term is given as $a = 6$.

2. Common ratio $r$: The common ratio is $r = 2$.

3. Number of terms $n$: The series runs from $n = 1$ to $n = 5$, so $n = 5$.

4. Substitute into the formula:

$S = \frac{6(1 - 2^5)}{1 - 2}$

Simplifying the terms:

$S = \frac{6(1 - 32)}{1 - 2} = \frac{6(-31)}{-1}$

$S = 186$

Thus, the value of $S$ is:

$S = 186$

Would you like more details, or do you have any questions on the solution?

Here are 5 related questions to expand your understanding:

1. What is a geometric series, and how does it differ from an arithmetic series?
2. How would the result change if the common ratio $r$ was negative?
3. Can this sum formula be applied to an infinite geometric series?
4. What happens to the sum if the number of terms $n$ is much larger?
5. How can you derive the sum formula for a geometric series?

Tip: Always check whether a geometric series is finite or infinite, as this impacts which formula you should use.