Prove by induction as it says. Please simple and detailed solution show each step Consider the Fibonacci numbers defined as follows: The first two Fibonacci numbers are 0 and 1,and each subsequent Fibonacci number is the sum of the two previous ones. The n-th Fibonaccinumber is denoted Fn. In other words, the Fibonacci numbers are defined recursively by therules:Fo := 0,F₁ := 1,F₁ = F₁-1 + Fi-2, for i > 2Fibonacci numbers come up naturally in several ways and are important in many applicationsbecause they have many, partially surprising properties such as the ones expressed in the fol-lowing theorems. E.g., the Fibonacci recurrence function is very useful for computing powersof matrices efficiently.Prove the following two properties about the Fibonacci numbers by induction:1. First we focus on the sequence of squared Fibonacci numbers.Theorem 1 (Sequence of Squared Fibonacci Numbers).Vn20: F+F+ +F²= FnFn+1.2. Given matrix multiplication defined as follows:(b11 b12)b21 b22,122) (621prove the following property by induction:Theorem 2. Vn> 1:a11a21(a₁b₁a12b21a11b12+a12b22)a21b11 + a22b21 a21b12+ a22b22,n(1 3)" = (Fr+¹FnFn+1 Fn

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Answered: 1. First we focus on the sequence of…

1. First we focus on the sequence of squared Fibonacci numbers. Theorem 1 (Sequence of Squared Fibonacci Numbers). Vn20: F+F²+ ... + F² = F₂Fn+1.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 91E

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Question

Prove by induction as it says. Please simple and detailed solution show each step

Consider the Fibonacci numbers defined as follows: The first two Fibonacci numbers are 0 and 1,
and each subsequent Fibonacci number is the sum of the two previous ones. The n-th Fibonacci
number is denoted Fn. In other words, the Fibonacci numbers are defined recursively by the
rules:
Fo := 0,
F₁ := 1,
F₁ = F₁-1 + Fi-2, for i > 2
Fibonacci numbers come up naturally in several ways and are important in many applications
because they have many, partially surprising properties such as the ones expressed in the fol-
lowing theorems. E.g., the Fibonacci recurrence function is very useful for computing powers
of matrices efficiently.
Prove the following two properties about the Fibonacci numbers by induction:
1. First we focus on the sequence of squared Fibonacci numbers.
Theorem 1 (Sequence of Squared Fibonacci Numbers).
Vn20: F+F+ +F²= FnFn+1.
2. Given matrix multiplication defined as follows:
(b11 b12)
b21 b22,
122) (621
prove the following property by induction:
Theorem 2. Vn> 1:
a11
a21
(a₁b₁a12b21a11b12+a12b22)
a21b11 + a22b21 a21b12+ a22b22,
n
(1 3)" = (Fr+¹
Fn
Fn+1 Fn

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