{tr(F")}~-1~ n=1 (9) Find a recurrence relation that generates the sequence tr
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 8RE
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Question
![**Fibonacci Numbers:**
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
**Definition:**
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:**
\[ F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5180d394-a6a1-427d-ae5d-e272ff18746e%2Fa297eb14-0c49-4125-9619-fd0004e263f9%2F17txm0l_processed.png&w=3840&q=75)
Transcribed Image Text:**Fibonacci Numbers:**
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
**Definition:**
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:**
\[ F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}. \]
Transcribed Image Text:**(9)** Find a recurrence relation that generates the sequence \(\left\{ \text{tr} \left( F^n \right) \right\}_{n=1}^{\infty}\).
**Explanation:**
In this problem, you are tasked with finding a recurrence relation that produces the sequence formed by the trace of powers of a matrix \( F \). The term \(\text{tr} \left( F^n \right)\) denotes the trace of the matrix \( F^n \), which is the sum of the elements on the main diagonal of \( F^n \).
A recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given; each subsequent term of the sequence is a function of the preceding terms. You need to express \(\text{tr} \left( F^n \right)\) based on previous terms like \(\text{tr} \left( F^{n-1} \right)\), \(\text{tr} \left( F^{n-2} \right)\), etc.
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