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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![**Geometric Sequence Problem**
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**Question 15:**
If 40, \( n \), and 10 form the first three terms of a geometric sequence, then which of the following is the value of \( n \)?
1. 16
2. 18
3. 20
4. 24
**Answer Choices:**
- ( ) 1
- ( ) 2
- ( ) 3
- ( ) 4
---
Explanation:
In a geometric sequence, each term after the first is the product of the previous term and a constant ratio \( r \). That is, the sequence can be represented as:
\[ a, ar, ar^2, ... \]
Given that 40, \( n \), and 10 form the first three terms of a geometric sequence:
- The first term \( a \) is 40.
- The second term \( n \) is \( ar \).
- The third term is 10.
Since the third term is \( ar^2 \), we have:
\[ ar^2 = 10 \]
From the first term:
\[ a = 40 \]
The second term:
\[ ar = n \]
\[ 40r = n \]
Substituting \( a = 40 \) into the equation \( ar^2 = 10 \):
\[ 40r^2 = 10 \]
\[ r^2 = \frac{10}{40} \]
\[ r^2 = \frac{1}{4} \]
\[ r = \frac{1}{2} \, \text{or} \, r = -\frac{1}{2} \]
If \( r = \frac{1}{2} \):
\[ n = 40 \times \frac{1}{2} = 20 \]
Therefore, \( n = 20 \), which corresponds to option 3.
Correct Answer: **3. 20**
Remember that identifying sequences and understanding their properties are crucial skills in mathematics that apply widely, from algebra to advanced calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1adaf557-0ebb-4539-a740-89b2306a05ed%2Fe817a69e-5542-4846-a339-7915eabf0b34%2Fvctnxz.jpeg&w=3840&q=75)
Transcribed Image Text:**Geometric Sequence Problem**
---
**Question 15:**
If 40, \( n \), and 10 form the first three terms of a geometric sequence, then which of the following is the value of \( n \)?
1. 16
2. 18
3. 20
4. 24
**Answer Choices:**
- ( ) 1
- ( ) 2
- ( ) 3
- ( ) 4
---
Explanation:
In a geometric sequence, each term after the first is the product of the previous term and a constant ratio \( r \). That is, the sequence can be represented as:
\[ a, ar, ar^2, ... \]
Given that 40, \( n \), and 10 form the first three terms of a geometric sequence:
- The first term \( a \) is 40.
- The second term \( n \) is \( ar \).
- The third term is 10.
Since the third term is \( ar^2 \), we have:
\[ ar^2 = 10 \]
From the first term:
\[ a = 40 \]
The second term:
\[ ar = n \]
\[ 40r = n \]
Substituting \( a = 40 \) into the equation \( ar^2 = 10 \):
\[ 40r^2 = 10 \]
\[ r^2 = \frac{10}{40} \]
\[ r^2 = \frac{1}{4} \]
\[ r = \frac{1}{2} \, \text{or} \, r = -\frac{1}{2} \]
If \( r = \frac{1}{2} \):
\[ n = 40 \times \frac{1}{2} = 20 \]
Therefore, \( n = 20 \), which corresponds to option 3.
Correct Answer: **3. 20**
Remember that identifying sequences and understanding their properties are crucial skills in mathematics that apply widely, from algebra to advanced calculus.
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