Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Determining the First 5 Terms of a Recursive Sequence
**Problem Statement:**
What are the first 5 terms of the sequence represented by:
\[ a_n = 3n - 2a_{n-1}, \]
where \( a_1 = 4 \)?
**Instructions:**
Show all work for full credit.
### Solution:
To find the first five terms of the sequence, we perform the following steps:
1. **Given:**
\[ a_1 = 4 \]
2. **For \( a_2 \):**
\[ a_2 = 3(2) - 2a_1 \]
Substitute \( a_1 = 4 \):
\[ a_2 = 3(2) - 2(4) \]
\[ a_2 = 6 - 8 \]
\[ a_2 = -2 \]
3. **For \( a_3 \):**
\[ a_3 = 3(3) - 2a_2 \]
Substitute \( a_2 = -2 \):
\[ a_3 = 3(3) - 2(-2) \]
\[ a_3 = 9 + 4 \]
\[ a_3 = 13 \]
4. **For \( a_4 \):**
\[ a_4 = 3(4) - 2a_3 \]
Substitute \( a_3 = 13 \):
\[ a_4 = 3(4) - 2(13) \]
\[ a_4 = 12 - 26 \]
\[ a_4 = -14 \]
5. **For \( a_5 \):**
\[ a_5 = 3(5) - 2a_4 \]
Substitute \( a_4 = -14 \):
\[ a_5 = 3(5) - 2(-14) \]
\[ a_5 = 15 + 28 \]
\[ a_5 = 43 \]
**Conclusion:**
The first five terms of the sequence are:
\[ a_1 = 4 \]
\[ a_2 = -2 \]
\[ a_3 = 13 \]
\[ a_4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8cf7e672-b05a-4acb-ad51-f9b37172b8f4%2Feb9c8a31-735e-44fc-9706-b475f9b3dddc%2F9llh78_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining the First 5 Terms of a Recursive Sequence
**Problem Statement:**
What are the first 5 terms of the sequence represented by:
\[ a_n = 3n - 2a_{n-1}, \]
where \( a_1 = 4 \)?
**Instructions:**
Show all work for full credit.
### Solution:
To find the first five terms of the sequence, we perform the following steps:
1. **Given:**
\[ a_1 = 4 \]
2. **For \( a_2 \):**
\[ a_2 = 3(2) - 2a_1 \]
Substitute \( a_1 = 4 \):
\[ a_2 = 3(2) - 2(4) \]
\[ a_2 = 6 - 8 \]
\[ a_2 = -2 \]
3. **For \( a_3 \):**
\[ a_3 = 3(3) - 2a_2 \]
Substitute \( a_2 = -2 \):
\[ a_3 = 3(3) - 2(-2) \]
\[ a_3 = 9 + 4 \]
\[ a_3 = 13 \]
4. **For \( a_4 \):**
\[ a_4 = 3(4) - 2a_3 \]
Substitute \( a_3 = 13 \):
\[ a_4 = 3(4) - 2(13) \]
\[ a_4 = 12 - 26 \]
\[ a_4 = -14 \]
5. **For \( a_5 \):**
\[ a_5 = 3(5) - 2a_4 \]
Substitute \( a_4 = -14 \):
\[ a_5 = 3(5) - 2(-14) \]
\[ a_5 = 15 + 28 \]
\[ a_5 = 43 \]
**Conclusion:**
The first five terms of the sequence are:
\[ a_1 = 4 \]
\[ a_2 = -2 \]
\[ a_3 = 13 \]
\[ a_4
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