Answered: {tr(F")}~-1~ n=1 (9) Find a recurrence… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.4: Fractional Expressions

Problem 66E

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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:**

\[ F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}. \]

**(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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