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 3
Write an explicit formula for the geometric sequence:
\[ 4, \frac{4}{3}, \frac{4}{9}, \frac{4}{27}, \ldots \]
/(5 pts)
---
#### Explanation:
In this question, you are asked to write an explicit formula for the given geometric sequence. A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
1. **First Term (\( a \))**: The first term of the sequence is \( 4 \).
2. **Common Ratio (\( r \))**: To find the common ratio, divide the second term by the first term:
\[
r = \frac{\frac{4}{3}}{4} = \frac{4}{3} \times \frac{1}{4} = \frac{1}{3}
\]
The general formula for the \( n \)-th term of a geometric sequence is given by:
\[ a_n = a \cdot r^{n-1} \]
Using the values of \( a \) and \( r \) we found,
\[ a_n = 4 \cdot \left( \frac{1}{3} \right)^{n-1} \]
So, the explicit formula for the given geometric sequence is:
\[ a_n = 4 \cdot \left( \frac{1}{3} \right)^{n-1} \]
This formula can be used to calculate any term in the sequence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c4ebd1d-4aff-44b9-b4f8-9546f55a75a9%2F3f35280f-83e2-4e46-9736-2c4c2cf8e656%2Fsynmsru_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 3
Write an explicit formula for the geometric sequence:
\[ 4, \frac{4}{3}, \frac{4}{9}, \frac{4}{27}, \ldots \]
/(5 pts)
---
#### Explanation:
In this question, you are asked to write an explicit formula for the given geometric sequence. A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
1. **First Term (\( a \))**: The first term of the sequence is \( 4 \).
2. **Common Ratio (\( r \))**: To find the common ratio, divide the second term by the first term:
\[
r = \frac{\frac{4}{3}}{4} = \frac{4}{3} \times \frac{1}{4} = \frac{1}{3}
\]
The general formula for the \( n \)-th term of a geometric sequence is given by:
\[ a_n = a \cdot r^{n-1} \]
Using the values of \( a \) and \( r \) we found,
\[ a_n = 4 \cdot \left( \frac{1}{3} \right)^{n-1} \]
So, the explicit formula for the given geometric sequence is:
\[ a_n = 4 \cdot \left( \frac{1}{3} \right)^{n-1} \]
This formula can be used to calculate any term in the sequence.
Transcribed Image Text:**Question 4**
What is the 11th term of the geometric sequence -1.5, -3, -6, -12, ...? (5 pts)
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