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Define the Fibonacci numbers as Fo = 0, F₁ = 1 and for n ≥ 2, set Fn = Fn-1+Fn-2. Show that Fn+1Fn-1 − F2 = (-1)" for n ≥ 1 and use this to deduce that gcd(Fn, Fn-1) = 1 for all n> 1.

Define the Fibonacci numbers as Fo = 0, F₁ = 1 and for n ≥ 2, set Fn = Fn-1+Fn-2. Show that Fn+1Fn-1 − F2 = (-1)" for n ≥ 1 and use this to deduce that gcd(Fn, Fn-1) = 1 for all n> 1.

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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Define the Fibonacci numbers as Fo= 0, F₁ = 1 and for n ≥ 2, set Fn = Fn-1+Fn-2. Show that Fn+1Fn-1 − F2 = (-1)" for n ≥ 1 and use this to deduce that gcd(Fn, Fn-1) = 1 for all n ≥ 1.
Transcribed Image Text:Define the Fibonacci numbers as Fo= 0, F₁ = 1 and for n ≥ 2, set Fn = Fn-1+Fn-2. Show that Fn+1Fn-1 − F2 = (-1)" for n ≥ 1 and use this to deduce that gcd(Fn, Fn-1) = 1 for all n ≥ 1.
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