Show all steps to find the sum of the first 11 terms of the geometric series given. Round answers to the nearest hundredth, if necessary. 9 +6+4+... Paragraph BI UVA E O + v ...
Show all steps to find the sum of the first 11 terms of the geometric series given. Round answers to the nearest hundredth, if necessary. 9 +6+4+... Paragraph BI UVA E O + v ...
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
![**Finding the Sum of the First 11 Terms of a Geometric Series**
To find the sum of the first 11 terms of the given geometric series, follow these steps. Round your answers to the nearest hundredth, if necessary.
Given geometric series: \( 9 + 6 + 4 + \ldots \)
1. **Identify the First Term \( (a) \) and Common Ratio \( (r) \):**
- First term \( a = 9 \)
- Common ratio \( r \):
\[
r = \frac{{\text{second term}}}{{\text{first term}}} = \frac{6}{9} = \frac{2}{3}
\]
2. **Sum of the First \( n \) Terms of a Geometric Series:**
The sum \( S_n \) of the first \( n \) terms of a geometric series is given by:
\[
S_n = a \frac{{1 - r^n}}{{1 - r}}
\]
where:
- \( S_n \) = sum of the first \( n \) terms
- \( a \) = first term
- \( r \) = common ratio
- \( n \) = number of terms
3. **Plugging in the Values:**
For the first 11 terms:
- \( a = 9 \)
- \( r = \frac{2}{3} \)
- \( n = 11 \)
Therefore:
\[
S_{11} = 9 \frac{{1 - \left( \frac{2}{3} \right)^{11}}}{{1 - \frac{2}{3}}}
\]
4. **Calculate \( \left( \frac{2}{3} \right)^{11} \):**
\[
\left( \frac{2}{3} \right)^{11} \approx 0.0185
\]
5. **Calculate \( 1 - \left( \frac{2}{3} \right)^{11} \):**
\[
1 - 0.0185 = 0.9815
\]
6. **Calculate the Denominator \( 1 - \frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4fcdecb9-56df-4c9c-aece-60df7fbe5b7a%2Ffc51772e-e141-4aba-9a0f-b0d23d9d6653%2Fo8ahcz_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Sum of the First 11 Terms of a Geometric Series**
To find the sum of the first 11 terms of the given geometric series, follow these steps. Round your answers to the nearest hundredth, if necessary.
Given geometric series: \( 9 + 6 + 4 + \ldots \)
1. **Identify the First Term \( (a) \) and Common Ratio \( (r) \):**
- First term \( a = 9 \)
- Common ratio \( r \):
\[
r = \frac{{\text{second term}}}{{\text{first term}}} = \frac{6}{9} = \frac{2}{3}
\]
2. **Sum of the First \( n \) Terms of a Geometric Series:**
The sum \( S_n \) of the first \( n \) terms of a geometric series is given by:
\[
S_n = a \frac{{1 - r^n}}{{1 - r}}
\]
where:
- \( S_n \) = sum of the first \( n \) terms
- \( a \) = first term
- \( r \) = common ratio
- \( n \) = number of terms
3. **Plugging in the Values:**
For the first 11 terms:
- \( a = 9 \)
- \( r = \frac{2}{3} \)
- \( n = 11 \)
Therefore:
\[
S_{11} = 9 \frac{{1 - \left( \frac{2}{3} \right)^{11}}}{{1 - \frac{2}{3}}}
\]
4. **Calculate \( \left( \frac{2}{3} \right)^{11} \):**
\[
\left( \frac{2}{3} \right)^{11} \approx 0.0185
\]
5. **Calculate \( 1 - \left( \frac{2}{3} \right)^{11} \):**
\[
1 - 0.0185 = 0.9815
\]
6. **Calculate the Denominator \( 1 - \frac{
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