https://file.aiquickdraw.com/mathbot/file/1730474593944axzgva5q.jpg

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.

Identify a1, r, and n for the sum of the geometric series and find the sum.

Learn how to calculate the sum of a finite geometric series with the first term a1 = 1/2, common ratio r = 3/4, and number of terms n = 5. This example includes step-by-step calculations and applies the geometric series sum formula to arrive at the answer 269/512.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation