The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.... are defined by the recursion formula x_{n+1}=x_{n}+x_{n-1}xn+1=xn+xn−1 , whith x_{1}=x_{2}=1x1=x2=1 .Prove that (x_{n}, x_{n+1}\right)=1(xn,xn+1)=1 and that x_{n}=xn= (a^{n}-b^{n}\) /(a-b)(an−bn)/(a−b) , where a and bare the roots of the quadratic equation x^{2}-x-1=0x2−x−1=0 .

The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.... are defined by the recursion formula x_{n+1}=x_{n}+x_{n-1}xn+1=xn+xn−1 , whith x_{1}=x_{2}=1x1=x2=1 .Prove that (x_{n}, x_{n+1}\right)=1(xn,xn+1)=1 and that x_{n}=xn= (a^{n}-b^{n}\) /(a-b)(an−bn)/(a−b) , where a and bare the roots of the quadratic equation x^{2}-x-1=0x2−x−1=0 .

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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