Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Problem Statement:**
Find the infinite sum of the geometric sequence with
\( a = 4, \, r = \frac{2}{7} \) if it exists.
\[ S_{\infty} = \text{\_\_\_\_\_\_} \]
**Explanation:**
1. **Parameters:**
- The first term of the sequence, \( a \), is 4.
- The common ratio, \( r \), is \( \frac{2}{7} \).
2. **Infinite Geometric Series Sum Formula:**
The sum of an infinite geometric series is given by:
\[
S_{\infty} = \frac{a}{1 - r}
\]
provided that \( |r| < 1 \).
3. **Application:**
Since \( r = \frac{2}{7} \) and \( |r| < 1 \), the infinite sum exists.
4. **Calculation:**
Substituting the values into the formula:
\[
S_{\infty} = \frac{4}{1 - \frac{2}{7}} = \frac{4}{\frac{5}{7}} = 4 \times \frac{7}{5} = \frac{28}{5}
\]
Therefore, the infinite sum of the sequence is \( \frac{28}{5} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d4850fc-7d28-42a5-9c0d-dd66822a7446%2F504b5fb0-3df2-411c-a852-42516e8c22ef%2F5rvje5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the infinite sum of the geometric sequence with
\( a = 4, \, r = \frac{2}{7} \) if it exists.
\[ S_{\infty} = \text{\_\_\_\_\_\_} \]
**Explanation:**
1. **Parameters:**
- The first term of the sequence, \( a \), is 4.
- The common ratio, \( r \), is \( \frac{2}{7} \).
2. **Infinite Geometric Series Sum Formula:**
The sum of an infinite geometric series is given by:
\[
S_{\infty} = \frac{a}{1 - r}
\]
provided that \( |r| < 1 \).
3. **Application:**
Since \( r = \frac{2}{7} \) and \( |r| < 1 \), the infinite sum exists.
4. **Calculation:**
Substituting the values into the formula:
\[
S_{\infty} = \frac{4}{1 - \frac{2}{7}} = \frac{4}{\frac{5}{7}} = 4 \times \frac{7}{5} = \frac{28}{5}
\]
Therefore, the infinite sum of the sequence is \( \frac{28}{5} \).
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