32. Write the rule for the following arithmetic sequence. -15,-24,-33,-42, ... A. a = -9n-6 B. an 9n-6 C. a = -9n+6 D. an = 9n + 6
32. Write the rule for the following arithmetic sequence. -15,-24,-33,-42, ... A. a = -9n-6 B. an 9n-6 C. a = -9n+6 D. an = 9n + 6
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer #32
![### Question 32: Writing the Rule for an Arithmetic Sequence
#### Problem Statement:
Determine the rule for the given arithmetic sequence:
\-15, -24, -33, -42, ...
Select the correct formula from the following options:
**A.** \( a_n = -9n - 6 \)
**B.** \( a_n = 9n - 6 \)
**C.** \( a_n = -9n + 6 \)
**D.** \( a_n = 9n + 6 \)
### Detailed Analysis:
The sequence provided is an arithmetic sequence, which means the difference between consecutive terms is constant. To solve this, we first need to find this common difference and then use it to write the general rule for the sequence.
#### Steps to Solve:
1. Identify the common difference (\(d\)) in the sequence:
\[
d = \text{Second term} - \text{First term} = -24 - (-15) = -24 + 15 = -9
\]
Similarly, performing the same check between other consecutive terms will confirm this common difference:
\[
-33 - (-24) = -33 + 24 = -9
\]
\[
-42 - (-33) = -42 + 33 = -9
\]
2. The first term (\(a_1\)) of the sequence:
\[
a_1 = -15
\]
3. The general formula for the nth term of an arithmetic sequence is:
\[
a_n = a_1 + (n-1)d
\]
4. Substitute \(a_1\) and \(d\) into the formula:
\[
a_n = -15 + (n-1)(-9)
\]
Simplify the expression step-by-step:
\[
a_n = -15 - 9n + 9
\]
\[
a_n = -9n - 6
\]
#### Conclusion:
The correct formula for the given arithmetic sequence is:
\[
a_n = -9n - 6
\]
Hence, the correct option is:
**A.** \( a_n = -9n - 6 \)
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This content is suitable for educational websites aimed at enhancing students' understanding of arithmetic sequences and their](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fce351b-90e5-44a7-899d-c11bfcc32a90%2Fff777717-bfba-4f29-a336-1f8f2ead42ea%2F1spy2hc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 32: Writing the Rule for an Arithmetic Sequence
#### Problem Statement:
Determine the rule for the given arithmetic sequence:
\-15, -24, -33, -42, ...
Select the correct formula from the following options:
**A.** \( a_n = -9n - 6 \)
**B.** \( a_n = 9n - 6 \)
**C.** \( a_n = -9n + 6 \)
**D.** \( a_n = 9n + 6 \)
### Detailed Analysis:
The sequence provided is an arithmetic sequence, which means the difference between consecutive terms is constant. To solve this, we first need to find this common difference and then use it to write the general rule for the sequence.
#### Steps to Solve:
1. Identify the common difference (\(d\)) in the sequence:
\[
d = \text{Second term} - \text{First term} = -24 - (-15) = -24 + 15 = -9
\]
Similarly, performing the same check between other consecutive terms will confirm this common difference:
\[
-33 - (-24) = -33 + 24 = -9
\]
\[
-42 - (-33) = -42 + 33 = -9
\]
2. The first term (\(a_1\)) of the sequence:
\[
a_1 = -15
\]
3. The general formula for the nth term of an arithmetic sequence is:
\[
a_n = a_1 + (n-1)d
\]
4. Substitute \(a_1\) and \(d\) into the formula:
\[
a_n = -15 + (n-1)(-9)
\]
Simplify the expression step-by-step:
\[
a_n = -15 - 9n + 9
\]
\[
a_n = -9n - 6
\]
#### Conclusion:
The correct formula for the given arithmetic sequence is:
\[
a_n = -9n - 6
\]
Hence, the correct option is:
**A.** \( a_n = -9n - 6 \)
---
This content is suitable for educational websites aimed at enhancing students' understanding of arithmetic sequences and their
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