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The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1, whit x1 = 1.Prove that (xn, xn+1) = 1 and that xn (a" – b") /(a – b), where a and bare the roots of the quadratic equation x? – x – 1 = 0. X2 - -

The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1, whit x1 = 1.Prove that (xn, xn+1) = 1 and that xn (a" – b") /(a – b), where a and bare the roots of the quadratic equation x? – x – 1 = 0. X2 - -

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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The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13... are defined by the recursion formula xn+1 = Xn + Xn-1, whit X1 = 1.Prove that (xn, Xn+1) = 1 and that xn (a" – b") /(a – b), where a and bare the roots of the quadratic equation x2 – x – 1 = 0. X2 - - -
Transcribed Image Text:The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13... are defined by the recursion formula xn+1 = Xn + Xn-1, whit X1 = 1.Prove that (xn, Xn+1) = 1 and that xn (a" – b") /(a – b), where a and bare the roots of the quadratic equation x2 – x – 1 = 0. X2 - - -
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