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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 of the characteristic equation, we have: Tags fibonacci, roots, solution, unique Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mathmari Applied Math 2 April 11th, 2013 01:36 PM shine123 Linear Algebra 4 September 28th, 2012 08:00 AM watkd Algebra 4 July 26th, 2012 04:58 PM imbored Real Analysis 0 February 23rd, 2010 09:49 PM imbored Applied Math 0 December 31st, 1969 04:00 PM

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