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
Find the infinite sum of the geometric sequence with a = 2, r° =
4
- if it exists
![**Finding the Infinite Sum of a Geometric Sequence**
Given:
- Initial term (a): \(2\)
- Common ratio (r): \(\frac{4}{5}\)
We are required to find the infinite sum \(S_\infty\) of the geometric sequence if it exists.
The formula for the infinite sum of a geometric sequence \(S_\infty\) is given by:
\[ S_\infty = \frac{a}{1 - r} \]
where \(|r| < 1\) for the series to converge.
### Applying the Formula
1. **Identify the values:**
- \( a = 2 \)
- \( r = \frac{4}{5} \)
2. **Check the condition for convergence:**
- Since \(\left|r\right| = \left|\frac{4}{5}\right| = 0.8\), which is less than 1, the series converges.
3. **Substitute into the formula:**
\[ S_\infty = \frac{2}{1 - \frac{4}{5}} \]
4. **Compute the sum:**
\[ S_\infty = \frac{2}{\frac{1}{5}} \]
\[ S_\infty = 2 \times 5 \]
\[ S_\infty = 10 \]
Thus, the infinite sum \( S_\infty \) is \( \boxed{10} \).
*[Note: Insert a box for student input alongside the computed formula as shown in the original prompt to help with practicing calculations.]*
### Interface Elements
- **Add Work:** This button could allow students to show their step-by-step solution or computation to arrive at the sum.
- **Next Question:** This button could lead students to the subsequent problem for further practice.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f973d2e-ff2e-4b11-9052-66f4bc90f91c%2F4b2a1746-93f2-4e15-9efe-2cc7a0b90326%2F5krroil_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Infinite Sum of a Geometric Sequence**
Given:
- Initial term (a): \(2\)
- Common ratio (r): \(\frac{4}{5}\)
We are required to find the infinite sum \(S_\infty\) of the geometric sequence if it exists.
The formula for the infinite sum of a geometric sequence \(S_\infty\) is given by:
\[ S_\infty = \frac{a}{1 - r} \]
where \(|r| < 1\) for the series to converge.
### Applying the Formula
1. **Identify the values:**
- \( a = 2 \)
- \( r = \frac{4}{5} \)
2. **Check the condition for convergence:**
- Since \(\left|r\right| = \left|\frac{4}{5}\right| = 0.8\), which is less than 1, the series converges.
3. **Substitute into the formula:**
\[ S_\infty = \frac{2}{1 - \frac{4}{5}} \]
4. **Compute the sum:**
\[ S_\infty = \frac{2}{\frac{1}{5}} \]
\[ S_\infty = 2 \times 5 \]
\[ S_\infty = 10 \]
Thus, the infinite sum \( S_\infty \) is \( \boxed{10} \).
*[Note: Insert a box for student input alongside the computed formula as shown in the original prompt to help with practicing calculations.]*
### Interface Elements
- **Add Work:** This button could allow students to show their step-by-step solution or computation to arrive at the sum.
- **Next Question:** This button could lead students to the subsequent problem for further practice.
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