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
Find a1 and d please
![**Problem: Finding the First Term and Common Difference in an Arithmetic Series**
Given:
- The sum of the first 16 terms, \( S_{16} = 360 \).
- The 16th term, \( a_{16} = 45 \).
Task: Determine the first term (\( a_1 \)) and the common difference (\( d \)) of the arithmetic series.
### Steps to Solve:
1. **Formula for the Sum of an Arithmetic Series:**
\[
S_n = \frac{n}{2} (a_1 + a_n)
\]
Where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.
2. **Formula for the nth Term of an Arithmetic Series:**
\[
a_n = a_1 + (n-1)d
\]
3. **Use the Given Values:**
Using \( S_{16} = 360 \) and \( a_{16} = 45 \), substitute into the formulas to find \( a_1 \) and \( d \).
**Equations:**
- From the sum formula:
\[
360 = \frac{16}{2} (a_1 + 45)
\]
- Simplify and solve for \( a_1 \) in terms of \( d \):
\[
360 = 8(a_1 + 45)
\]
\[
45 = a_1 + 15d
\]
**Solutions:**
1. Solve the system of equations obtained from the above steps to find \( a_1 \) and \( d \).
2. Substitute these values into either original formula to verify the solution.
**Calculation results:**
- Find \( a_1 \) first by simplifying from the sum equation.
- Use the nth term formula to find \( d \).
Once solved, you can fill in the values for \( a_1 \) and \( d \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F228397b8-cd0e-4924-bc62-b0b7f3293021%2Fba338d6f-c61b-4dce-a457-ff38f52acc24%2F1t9w3bp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Finding the First Term and Common Difference in an Arithmetic Series**
Given:
- The sum of the first 16 terms, \( S_{16} = 360 \).
- The 16th term, \( a_{16} = 45 \).
Task: Determine the first term (\( a_1 \)) and the common difference (\( d \)) of the arithmetic series.
### Steps to Solve:
1. **Formula for the Sum of an Arithmetic Series:**
\[
S_n = \frac{n}{2} (a_1 + a_n)
\]
Where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.
2. **Formula for the nth Term of an Arithmetic Series:**
\[
a_n = a_1 + (n-1)d
\]
3. **Use the Given Values:**
Using \( S_{16} = 360 \) and \( a_{16} = 45 \), substitute into the formulas to find \( a_1 \) and \( d \).
**Equations:**
- From the sum formula:
\[
360 = \frac{16}{2} (a_1 + 45)
\]
- Simplify and solve for \( a_1 \) in terms of \( d \):
\[
360 = 8(a_1 + 45)
\]
\[
45 = a_1 + 15d
\]
**Solutions:**
1. Solve the system of equations obtained from the above steps to find \( a_1 \) and \( d \).
2. Substitute these values into either original formula to verify the solution.
**Calculation results:**
- Find \( a_1 \) first by simplifying from the sum equation.
- Use the nth term formula to find \( d \).
Once solved, you can fill in the values for \( a_1 \) and \( d \).
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