5. Use polynomial fitting to find a closed formula for the sequence (ªn)n € N₁: 1,3, 11, 31, 69, ...
5. Use polynomial fitting to find a closed formula for the sequence (ªn)n € N₁: 1,3, 11, 31, 69, ...
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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Question
![**Problem 5:**
Use polynomial fitting to find a closed formula for the sequence \((a_n)_{n \in \mathbb{N}_0}\):
\[1, 3, 11, 31, 69, \ldots\]
**Explanation:**
In this problem, you are required to use polynomial fitting techniques to determine a closed formula for the given sequence of numbers. The sequence is \(1, 3, 11, 31, 69, \ldots\). This involves finding a polynomial expression \(P(n)\) such that its values correspond to the terms of the sequence for integer values of \(n\).
### Steps to Solve:
1. **Identify the Pattern in Differences:**
- Begin by calculating the differences between consecutive terms of the sequence to identify if there's a consistent pattern that could indicate the degree of the polynomial.
2. **Polynomial Assumption:**
- Assume a polynomial of degree \(d\), and establish equations using the initial terms to solve for the coefficients of the polynomial.
3. **Solve the System of Equations:**
- Use methods such as substitution or matrix operations to find the coefficients.
4. **Verification:**
- Verify the found polynomial expression with additional terms in the sequence to ensure its correctness.
Understanding and applying these steps effectively will enable you to derive an accurate closed-form expression for the sequence. This skill is crucial in mathematical analysis and computer science for sequences and series analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5867f8dc-778e-4b8e-9045-c957ac9725cc%2F4ae08629-b273-412b-a20e-f593c3063644%2F54o646i_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 5:**
Use polynomial fitting to find a closed formula for the sequence \((a_n)_{n \in \mathbb{N}_0}\):
\[1, 3, 11, 31, 69, \ldots\]
**Explanation:**
In this problem, you are required to use polynomial fitting techniques to determine a closed formula for the given sequence of numbers. The sequence is \(1, 3, 11, 31, 69, \ldots\). This involves finding a polynomial expression \(P(n)\) such that its values correspond to the terms of the sequence for integer values of \(n\).
### Steps to Solve:
1. **Identify the Pattern in Differences:**
- Begin by calculating the differences between consecutive terms of the sequence to identify if there's a consistent pattern that could indicate the degree of the polynomial.
2. **Polynomial Assumption:**
- Assume a polynomial of degree \(d\), and establish equations using the initial terms to solve for the coefficients of the polynomial.
3. **Solve the System of Equations:**
- Use methods such as substitution or matrix operations to find the coefficients.
4. **Verification:**
- Verify the found polynomial expression with additional terms in the sequence to ensure its correctness.
Understanding and applying these steps effectively will enable you to derive an accurate closed-form expression for the sequence. This skill is crucial in mathematical analysis and computer science for sequences and series analysis.
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