- #1

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Where Phi is the Golden ratio.

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- Thread starter billy2908
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- #1

- 12

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Where Phi is the Golden ratio.

- #2

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Just use the formula for nth fibonacci number.

- #3

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- #4

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Remember that the nth Fibonacci number is:

[tex]F_n = \frac {1} {\sqrt{5}} \left [ \phi^n - \left ( \frac {-1}{\phi}\right)^n\right][/tex]

and you can multiply F_{n}^{2} through your equation:

[tex] \left | F_{n+1} \times F_{n} - \phi \times F^2_n \right | < \frac {1}{2} [/tex]

What happens when you set the whole equation to terms of [itex]\phi[/itex]?

Note that for the first*3* Fibonacci numbers: 0,1,1 the formula doesn't work. Additionally, as we get deeper into the Fibonacci sequence, [itex]F_n[/itex] as [itex]n \to \infty[/itex], what does [itex] \left | F_{n+1} \times F_{n} - \phi \times F^2_n \right |[/itex] approach? *hint: you might see the value first for a specific very low n*

[tex]F_n = \frac {1} {\sqrt{5}} \left [ \phi^n - \left ( \frac {-1}{\phi}\right)^n\right][/tex]

and you can multiply F

[tex] \left | F_{n+1} \times F_{n} - \phi \times F^2_n \right | < \frac {1}{2} [/tex]

What happens when you set the whole equation to terms of [itex]\phi[/itex]?

Note that for the first

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