To find the sum of a finite geometric series, we can use the formula:

$S_n = a_1 \frac{1 - r^n}{1 - r}$

where:

• $a_1 = \frac{1}{2}$
• $r = \frac{3}{4}$
• $n = 5$

Substituting these values into the formula:

$S_5 = \frac{1}{2} \cdot \frac{1 - \left( \frac{3}{4} \right)^5}{1 - \frac{3}{4}}$

Calculating each part:

1. Calculate $\left( \frac{3}{4} \right)^5$: $\left( \frac{3}{4} \right)^5 = \frac{243}{1024}$

2. Find $1 - \frac{243}{1024}$: $1 - \frac{243}{1024} = \frac{1024 - 243}{1024} = \frac{781}{1024}$

3. Calculate the denominator $1 - \frac{3}{4} = \frac{1}{4}$.

4. Compute the sum $S_5$: $S_5 = \frac{1}{2} \cdot \frac{781}{1024} \cdot \frac{4}{1} = \frac{781 \times 2}{1024} = \frac{1562}{1024} = \frac{269}{512}$

The correct answer is:

$\frac{269}{512}$

Would you like a more detailed explanation, or do you have any questions?

Here are five related questions to expand on this:

1. How is the common ratio $r$ in a geometric series determined?
2. What would change in the sum if $r > 1$?
3. How do you find the sum of an infinite geometric series?
4. What happens if the series starts with a negative first term?
5. How would the formula for the sum change if $n$ were a large number?

Tip: Always double-check calculations in each step when dealing with fractions in geometric series to avoid small errors.