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 November 13th, 2012, 04:15 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(n-1) 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, 09:31 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 n-th 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, 09:58 PM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Unique solution to roots of Fibonacci ? As a follow-up (to the OP, not you Charles ), can you prove that for either root $r$ of the characteristic equation, we have: $r^n=F_{n-1}+F_{n}r$

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