# Differences

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+ | ====== Fibonacci Numbers ====== | ||

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+ | **Fibonacci numbers** are defined by the following recurrance: | ||

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+ | $ F_0\ =\ 0,$\\ | ||

+ | $ F_1\ =\ 1,$\\ | ||

+ | $F_i\ =\ F_{ i-1 } +\ F_{ i-2 }\ for\ i\geq 2.$ | ||

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+ | Each fibonacci number is the sum of the two previous ones, yielding the sequence | ||

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+ | $0,1,1,2,3,5,8,13,21,34,55, ...$ | ||

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+ | Fibonacci numbers are related to the **golden ratio** $\phi$ and to its | ||

+ | conjugate $\hat{\phi}$ which are the two roots of the equation: | ||

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+ | $x^2\ =\ x\ +\ 1$ | ||

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+ | and are given by the following: | ||

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+ | $ \phi = \frac{1+\sqrt{ 5 }}{2}$\\ | ||

+ | $ \hat{ \phi } = \frac{1-\sqrt{ 5 }}{2} $ | ||

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+ | Where we have: | ||

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+ | $F_i = \frac{\phi ^i-\hat{\phi}^i}{\sqrt{5}}$ | ||

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+ | Fibonacci numbers grow exponentially. | ||

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