Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 1SE: What is an nth partial sum?
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In Exercises 11 and 12, write the first five terms of the recursively defined sequence.
![### Problem Statement: Sequence Defined by Recurrence Relation
**Problem 12.** Consider the sequence defined by the recurrence relation \( a_{k+1} = \frac{1}{3}a_k^2 \) with the initial term \( a_1 = 6 \).
In this problem, the sequence starts with \( a_1 \) equal to 6. Each subsequent term in the sequence \( a_{k+1} \) is obtained by squaring the current term \( a_k \) and then dividing by 3.
**Key Concepts:**
- **Recurrence Relation**: A recurrence relation is an equation that recursively defines a sequence, where each term is a function of its preceding terms.
- **Initial Term**: The initial term \( a_1 \) is the first term of the sequence, which is given as 6 in this problem.
**Example Calculation**:
Given the initial term \( a_1 = 6 \), we can compute the next term in the sequence (\( a_2 \)) as follows:
\[ a_2 = \frac{1}{3} a_1^2 = \frac{1}{3} \times 6^2 = \frac{1}{3} \times 36 = 12 \]
Continuing this process, we can find the next few terms in the sequence.
### Steps to Solve:
1. **Start with the initial term**:
\[ a_1 = 6 \]
2. **Use the recurrence relation to find subsequent terms**:
\[ a_{k+1} = \frac{1}{3} a_k^2 \]
Calculate \( a_2 \):
\[ a_2 = \frac{1}{3} \times 6^2 = 12 \]
Calculate \( a_3 \):
\[ a_3 = \frac{1}{3} \times 12^2 = 48 \]
Calculate \( a_4 \):
\[ a_4 = \frac{1}{3} \times 48^2 = 768 \]
3. **Continue this process for as many terms as needed**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbfd0029-e037-4bef-a6dc-2fab8c35ba69%2F091c88da-31d3-4d44-99c2-b92317564c48%2Fvbjkuh7.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement: Sequence Defined by Recurrence Relation
**Problem 12.** Consider the sequence defined by the recurrence relation \( a_{k+1} = \frac{1}{3}a_k^2 \) with the initial term \( a_1 = 6 \).
In this problem, the sequence starts with \( a_1 \) equal to 6. Each subsequent term in the sequence \( a_{k+1} \) is obtained by squaring the current term \( a_k \) and then dividing by 3.
**Key Concepts:**
- **Recurrence Relation**: A recurrence relation is an equation that recursively defines a sequence, where each term is a function of its preceding terms.
- **Initial Term**: The initial term \( a_1 \) is the first term of the sequence, which is given as 6 in this problem.
**Example Calculation**:
Given the initial term \( a_1 = 6 \), we can compute the next term in the sequence (\( a_2 \)) as follows:
\[ a_2 = \frac{1}{3} a_1^2 = \frac{1}{3} \times 6^2 = \frac{1}{3} \times 36 = 12 \]
Continuing this process, we can find the next few terms in the sequence.
### Steps to Solve:
1. **Start with the initial term**:
\[ a_1 = 6 \]
2. **Use the recurrence relation to find subsequent terms**:
\[ a_{k+1} = \frac{1}{3} a_k^2 \]
Calculate \( a_2 \):
\[ a_2 = \frac{1}{3} \times 6^2 = 12 \]
Calculate \( a_3 \):
\[ a_3 = \frac{1}{3} \times 12^2 = 48 \]
Calculate \( a_4 \):
\[ a_4 = \frac{1}{3} \times 48^2 = 768 \]
3. **Continue this process for as many terms as needed**.
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