Chapter8: Sequences, Series,and Probability
Section8.2: Arithmetic Sequences And Partial Sums
Problem 1ECP: Write the first four terms of the arithmetic sequence whose nth term is 3n1. Then find the common...
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![### Problem: Predict the nth Term of the Sequence
Given the sequence:
\[ 3, 6, 9, 12, 15, \ldots \]
#### Question:
Predict the general, or nth term, \(a_n\) of the sequence.
#### Options:
- A. \( 3n \)
- B. \( n + 3 \)
- C. \( n^3 \)
- D. \( 4n \)
### Analysis:
The sequence given is \(3, 6, 9, 12, 15, \ldots \). Observing the sequence, we can see that each term increases by 3.
#### Steps to Determine the General Term:
1. **Identify the Pattern:**
- The first term (\(a_1\)) is 3.
- The second term (\(a_2\)) is 6.
- The third term (\(a_3\)) is 9.
- The fourth term (\(a_4\)) is 12.
- The fifth term (\(a_5\)) is 15.
2. **Calculate the Differences:**
- \(a_2 - a_1 = 6 - 3 = 3\)
- \(a_3 - a_2 = 9 - 6 = 3\)
- \(a_4 - a_3 = 12 - 9 = 3\)
- \(a_5 - a_4 = 15 - 12 = 3\)
The common difference is 3, indicating an arithmetic sequence.
3. **Determine the General Term Formula:**
- For an arithmetic sequence, the nth term can be expressed as:
\[
a_n = a_1 + (n-1)d
\]
Where \(a_1\) is the first term and \(d\) is the common difference.
4. **Substitute Values:**
- \(a_1 = 3\)
- \(d = 3\)
Therefore,
\[
a_n = 3 + (n-1) \cdot 3
\]
Simplifying,
\[
a_n = 3n
\]
#### Conclusion:
The general, or nth term, \(a_n\) of the sequence is:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F227fd297-25c6-41a3-8d4d-df25e1639e4d%2F749c4e6d-6b0c-4c58-b476-06635e624095%2Fpxhutkq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem: Predict the nth Term of the Sequence
Given the sequence:
\[ 3, 6, 9, 12, 15, \ldots \]
#### Question:
Predict the general, or nth term, \(a_n\) of the sequence.
#### Options:
- A. \( 3n \)
- B. \( n + 3 \)
- C. \( n^3 \)
- D. \( 4n \)
### Analysis:
The sequence given is \(3, 6, 9, 12, 15, \ldots \). Observing the sequence, we can see that each term increases by 3.
#### Steps to Determine the General Term:
1. **Identify the Pattern:**
- The first term (\(a_1\)) is 3.
- The second term (\(a_2\)) is 6.
- The third term (\(a_3\)) is 9.
- The fourth term (\(a_4\)) is 12.
- The fifth term (\(a_5\)) is 15.
2. **Calculate the Differences:**
- \(a_2 - a_1 = 6 - 3 = 3\)
- \(a_3 - a_2 = 9 - 6 = 3\)
- \(a_4 - a_3 = 12 - 9 = 3\)
- \(a_5 - a_4 = 15 - 12 = 3\)
The common difference is 3, indicating an arithmetic sequence.
3. **Determine the General Term Formula:**
- For an arithmetic sequence, the nth term can be expressed as:
\[
a_n = a_1 + (n-1)d
\]
Where \(a_1\) is the first term and \(d\) is the common difference.
4. **Substitute Values:**
- \(a_1 = 3\)
- \(d = 3\)
Therefore,
\[
a_n = 3 + (n-1) \cdot 3
\]
Simplifying,
\[
a_n = 3n
\]
#### Conclusion:
The general, or nth term, \(a_n\) of the sequence is:
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