My Math Forum Unique solution to roots of Fibonacci ?

 Number Theory Number Theory Math Forum

 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: 520 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: $r^n=F_{n-1}+F_{n}r$

 Tags fibonacci, roots, solution, unique

 Thread Tools Display Modes Linear 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

 Contact - Home - Forums - Cryptocurrency Forum - Top