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{tr (F")}"_? n=1 What kind the relation of this sequence to the Fibonacci sequence? (10) What is the famous name of the sequence trF"

{tr (F")}"_? n=1 What kind the relation of this sequence to the Fibonacci sequence? (10) What is the famous name of the sequence trF"

Advanced Engineering Mathematics
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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### Text Transcription **Question (10):** What is the famous name of the sequence \( \left\{ \text{tr} \left( F^n \right) \right\}_{n=1}^{\infty} \)? **Follow-up Question:** What kind of relation does this sequence have to the Fibonacci sequence? ### Explanation This question discusses a mathematical sequence derived from the trace of powers of a matrix \( F \). The task is to identify the well-known name of this sequence and explore its connection to the Fibonacci sequence. **Key Terms:** - **Sequence:** An ordered list of numbers that typically follows a specific rule or formula. - **Trace (tr):** The sum of the diagonal elements of a square matrix. - **Matrix \( F \):** A mathematical representation often used to solve linear algebra problems or represent linear transformations. This sequence is related to the Fibonacci sequence through matrix representations or transformations that yield Fibonacci numbers when manipulated in specified ways.
Transcribed Image Text:### Text Transcription **Question (10):** What is the famous name of the sequence \( \left\{ \text{tr} \left( F^n \right) \right\}_{n=1}^{\infty} \)? **Follow-up Question:** What kind of relation does this sequence have to the Fibonacci sequence? ### Explanation This question discusses a mathematical sequence derived from the trace of powers of a matrix \( F \). The task is to identify the well-known name of this sequence and explore its connection to the Fibonacci sequence. **Key Terms:** - **Sequence:** An ordered list of numbers that typically follows a specific rule or formula. - **Trace (tr):** The sum of the diagonal elements of a square matrix. - **Matrix \( F \):** A mathematical representation often used to solve linear algebra problems or represent linear transformations. This sequence is related to the Fibonacci sequence through matrix representations or transformations that yield Fibonacci numbers when manipulated in specified ways.
Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Define the Fibonacci sequence \(\{f_n\}_{n=0}^\infty\) by a recurrence relation (2nd order linear difference equation): \[ f_{n+2} = f_{n+1} + f_n, \quad n \geq 0, \quad f_0 = 0, \quad f_1 = 1. \] Fibonacci matrix: \( F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}. \) ### Explanation: - **Fibonacci Sequence:** A sequence of numbers where each number is the sum of the two preceding ones, often starting with 0 and 1. - **Recurrence Relation:** Describes how each term in the sequence is derived from the previous terms. - **Fibonacci Matrix:** A 2x2 matrix used in linear algebra to represent the recurrence relation in matrix form, which helps compute Fibonacci numbers efficiently.
Transcribed Image Text:Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Define the Fibonacci sequence \(\{f_n\}_{n=0}^\infty\) by a recurrence relation (2nd order linear difference equation): \[ f_{n+2} = f_{n+1} + f_n, \quad n \geq 0, \quad f_0 = 0, \quad f_1 = 1. \] Fibonacci matrix: \( F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}. \) ### Explanation: - **Fibonacci Sequence:** A sequence of numbers where each number is the sum of the two preceding ones, often starting with 0 and 1. - **Recurrence Relation:** Describes how each term in the sequence is derived from the previous terms. - **Fibonacci Matrix:** A 2x2 matrix used in linear algebra to represent the recurrence relation in matrix form, which helps compute Fibonacci numbers efficiently.
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