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...
Related questions
Question
![**Problem Statement:**
Find the nth term of an arithmetic sequence given \( a_4 = 34 \) and \( a_{10} = 52 \).
**Explanation for Educational Context:**
To solve this problem, you need to find the common difference and the general formula for the nth term of the arithmetic sequence. An arithmetic sequence can be defined by the formula:
\[ a_n = a_1 + (n-1) \cdot d \]
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
1. **Find the Common Difference (d):**
Using the terms provided:
\( a_4 = a_1 + 3d = 34 \)
\( a_{10} = a_1 + 9d = 52 \)
Subtract the first equation from the second:
\[(a_1 + 9d) - (a_1 + 3d) = 52 - 34\]
\[6d = 18\]
\[d = 3\]
2. **Find the First Term (a1):**
Substitute \( d = 3 \) back into the equation for \( a_4 \):
\[a_1 + 3 \times 3 = 34\]
\[a_1 + 9 = 34\]
\[a_1 = 25\]
3. **General Formula for the nth Term:**
Substitute \( a_1 = 25 \) and \( d = 3 \) back into the general formula:
\[a_n = 25 + (n-1) \cdot 3\]
\[a_n = 25 + 3n - 3\]
\[a_n = 3n + 22\]
Thus, the nth term of the arithmetic sequence is given by:
\[a_n = 3n + 22\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59fae379-c2a2-4553-824b-3c3ac08e75aa%2Fd95884a8-a16a-480f-969a-648ee50f7061%2Ft0f2io6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the nth term of an arithmetic sequence given \( a_4 = 34 \) and \( a_{10} = 52 \).
**Explanation for Educational Context:**
To solve this problem, you need to find the common difference and the general formula for the nth term of the arithmetic sequence. An arithmetic sequence can be defined by the formula:
\[ a_n = a_1 + (n-1) \cdot d \]
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
1. **Find the Common Difference (d):**
Using the terms provided:
\( a_4 = a_1 + 3d = 34 \)
\( a_{10} = a_1 + 9d = 52 \)
Subtract the first equation from the second:
\[(a_1 + 9d) - (a_1 + 3d) = 52 - 34\]
\[6d = 18\]
\[d = 3\]
2. **Find the First Term (a1):**
Substitute \( d = 3 \) back into the equation for \( a_4 \):
\[a_1 + 3 \times 3 = 34\]
\[a_1 + 9 = 34\]
\[a_1 = 25\]
3. **General Formula for the nth Term:**
Substitute \( a_1 = 25 \) and \( d = 3 \) back into the general formula:
\[a_n = 25 + (n-1) \cdot 3\]
\[a_n = 25 + 3n - 3\]
\[a_n = 3n + 22\]
Thus, the nth term of the arithmetic sequence is given by:
\[a_n = 3n + 22\]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning