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November 13th, 2012, 05:15 PM   #1
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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.
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November 13th, 2012, 10:31 PM   #2
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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.
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November 13th, 2012, 10:58 PM   #3
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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:

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