Recall the definition of Fibonacci numbers:• F1 = 0• F2 = 1• Fn = Fn−1 + Fn−2For example, the first Fibonacci numbers are 0,1,1,2,3,5,8,13,21 ...Show that the following formula is valid ∀n ∈N:ImageHINT1: Using a direct calculation, first prove that φ2 = 1 + φ andSimilarly, (1 −φ) 2=1 + (1 −φ).HINT2: Uses induction. In hip. of induction assumed true for n and forn −1 In the inductive step calculate Fn + 1 = Fn + Fn − 1 and use the HINT1. pn-1 - (1– )"-1V5F, =Donde:1+ V5= み2

Given Fn=φn-1-(1-φ)n-15, where φ=1+52 So Fn=φn-1-(1-φ)n-15=151+52n-1-1-52n-1…

Answered: Recall the definition of Fibonacci…

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

Recall the definition of Fibonacci numbers:
• F₁ = 0
• F₂ = 1
• F_n = F_n−1 + F_n−2

For example, the first Fibonacci numbers are 0,1,1,2,3,5,8,13,21 ...

Show that the following formula is valid ∀n ∈N:

Image

HINT1: Using a direct calculation, first prove that φ² = 1 + φ and
Similarly, (1 −φ) ²=1 + (1 −φ).
HINT2: Uses induction. In hip. of induction assumed true for n and for
n −1 In the inductive step calculate F_{n + 1} = F_n + F_{n − 1} and use the HINT1.