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,...
Related questions
Question
![**Problem Statement:**
Find the sum of the following infinite geometric series, or state that it is not possible.
\[
\sum_{k=1}^{\infty} 8 \left(\frac{-3}{4}\right)^k
\]
_To solve this problem:_
1. **Identify the First Term (a) and Common Ratio (r):**
- The first term (a) can be identified as the first term of the series when \(k=1\).
- The common ratio (r) is the factor by which each term of the series is multiplied to get the next term.
2. **Geometric Series Sum Formula for |r| < 1:**
- For an infinite geometric series to have a sum, the absolute value of the common ratio |r| must be less than 1.
- The sum \(S\) of an infinite geometric series is given by:
\[
S = \frac{a}{1 - r}
\]
3. **Apply the Values:**
- Here, the first term \(a_1 = 8 \left(\frac{-3}{4}\right)^1 = 8 \times \frac{-3}{4} = -6\)
- The common ratio \(r = \frac{-3}{4}\)
4. **Check if |r| < 1:**
- Here, \(|\frac{-3}{4}| = \frac{3}{4} < 1\)
5. **Calculate the Sum:**
- Substitute the values into the formula:
\[
S = \frac{a}{1 - r} = \frac{-6}{1 - \left(\frac{-3}{4}\right)} = \frac{-6}{1 + \frac{3}{4}} = \frac{-6}{\frac{7}{4}} = -6 \times \frac{4}{7} = \frac{-24}{7}
\]
Thus, the sum of the given infinite geometric series is:
\[
\boxed{\frac{-24}{7}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F585970c1-8db0-450f-987b-10f4de58f48c%2F73376a32-f681-456d-95a3-201090dccbc7%2Fcsvbc32_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the sum of the following infinite geometric series, or state that it is not possible.
\[
\sum_{k=1}^{\infty} 8 \left(\frac{-3}{4}\right)^k
\]
_To solve this problem:_
1. **Identify the First Term (a) and Common Ratio (r):**
- The first term (a) can be identified as the first term of the series when \(k=1\).
- The common ratio (r) is the factor by which each term of the series is multiplied to get the next term.
2. **Geometric Series Sum Formula for |r| < 1:**
- For an infinite geometric series to have a sum, the absolute value of the common ratio |r| must be less than 1.
- The sum \(S\) of an infinite geometric series is given by:
\[
S = \frac{a}{1 - r}
\]
3. **Apply the Values:**
- Here, the first term \(a_1 = 8 \left(\frac{-3}{4}\right)^1 = 8 \times \frac{-3}{4} = -6\)
- The common ratio \(r = \frac{-3}{4}\)
4. **Check if |r| < 1:**
- Here, \(|\frac{-3}{4}| = \frac{3}{4} < 1\)
5. **Calculate the Sum:**
- Substitute the values into the formula:
\[
S = \frac{a}{1 - r} = \frac{-6}{1 - \left(\frac{-3}{4}\right)} = \frac{-6}{1 + \frac{3}{4}} = \frac{-6}{\frac{7}{4}} = -6 \times \frac{4}{7} = \frac{-24}{7}
\]
Thus, the sum of the given infinite geometric series is:
\[
\boxed{\frac{-24}{7}}
\]
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