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Fibonacci Numbers and Golden Rule

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Show that |(frac{f_(n+1)}{f_n}) - phi| = frac{1}{(f_n)(phi^{n+1})}

and lim_{n --> infty} frac{f_{n+1}}{f_n} = phi,

where phi is the Golden Ration and is the unique positive root of phi^2 - phi - 1 = 0.

For some discussion on this question, see http://math.stackexchange.com/questions/106049/another-way-to-go-about-proving-the-limit-of-fibonaccis-sequence-quotient?lq=1

For a power series involving the nth fibonacci number f_n: Define f(z) = sum_{n = 0}^infty (f_n)(z^n). Show the the power series has a radius of convergence R = frac{1}{phi} and for |z| < R, f(z) = frac{1}{1 - z - z^2}

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Solution Summary

The expert examines Fibonacci Numbers and the Golden Rule.

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The recurrence formula of Fibonacci sequences can be written as:
f_n+1 - f_n - f_n-1 = 0

Searching for solutions of form
f_n = C*r^n ...

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