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November 13th, 2012, 05:15 PM  #1 
Newbie Joined: Oct 2012 Posts: 9 Thanks: 0  Unique solution to roots of Fibonacci ?
Hello, I am wondering if the solution to Fibonacci nth term solution has a unique solution ? U(n) = r^n U(n+1) = U(n) + U(n1) substituting r^n = U(n) gives me r^2 = r+1 Therefore r = [1 + sqrt(5)]/2 or r = [1  sqrt(5)]/2 Are these the only two unique solutions for r in recursive Fibonacci series ? Thanks for your help. 
November 13th, 2012, 10:31 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Unique solution to roots of Fibonacci ?
You can use those two numbers, phi and the inverse of phi, to generate the Fibonacci numbers. In particular, F_n = (phi^n  (1/phi)^n)/sqrt(5)  off the top of my head. Since 1/phi < 1, the second term very nearly vanishes so the nth Fibonacci number is just about phi^n/sqrt(5). This means that the ratio of consecutive Fibonacci numbers is just about phi.

November 13th, 2012, 10:58 PM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Unique solution to roots of Fibonacci ?
As a followup (to the OP, not you Charles ), can you prove that for either root of the characteristic equation, we have: 

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