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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Question
7
![**Problem Statement:**
Find the sum of the first 8 terms for the following arithmetic sequence.
Given:
- \( a_2 = 9 \)
- \( a_4 = 19 \)
**Question:**
The sum of the first 8 terms is [ ].
**Explanation:**
To solve this problem, follow these steps:
1. **Identify the common difference**:
- Use the formula of the n-th term of an arithmetic sequence, which is \( a_n = a_1 + (n-1)d \).
- Here, \( a_2 = 9 \) and \( a_4 = 19 \).
- Set up the equations:
- \( a_2 = a_1 + d = 9 \)
- \( a_4 = a_1 + 3d = 19 \)
- Solve these equations simultaneously to find \( a_1 \) and \( d \).
2. **Calculate the sum of the first 8 terms**:
- The formula for the sum of the first n terms of an arithmetic sequence is:
\[
S_n = \frac{n}{2} \times (2a_1 + (n-1)d)
\]
- Substitute \( n = 8 \), along with the values for \( a_1 \) and \( d \), to find \( S_8 \).
This approach will yield the required sum for the first 8 terms of the sequence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F228397b8-cd0e-4924-bc62-b0b7f3293021%2Fe600c6e8-b48f-4279-8dc3-2d9180398581%2F4zktkzm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the sum of the first 8 terms for the following arithmetic sequence.
Given:
- \( a_2 = 9 \)
- \( a_4 = 19 \)
**Question:**
The sum of the first 8 terms is [ ].
**Explanation:**
To solve this problem, follow these steps:
1. **Identify the common difference**:
- Use the formula of the n-th term of an arithmetic sequence, which is \( a_n = a_1 + (n-1)d \).
- Here, \( a_2 = 9 \) and \( a_4 = 19 \).
- Set up the equations:
- \( a_2 = a_1 + d = 9 \)
- \( a_4 = a_1 + 3d = 19 \)
- Solve these equations simultaneously to find \( a_1 \) and \( d \).
2. **Calculate the sum of the first 8 terms**:
- The formula for the sum of the first n terms of an arithmetic sequence is:
\[
S_n = \frac{n}{2} \times (2a_1 + (n-1)d)
\]
- Substitute \( n = 8 \), along with the values for \( a_1 \) and \( d \), to find \( S_8 \).
This approach will yield the required sum for the first 8 terms of the sequence.
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