Need help w/ Fibonnaci and Golden Ratio proof

billy2908

Let F_n and F_n+1 be successive Fibonnaci numbers. Then |(F_n+1)/(F_n) - Phi | < 1/(2(F_n)^2)

Where Phi is the Golden ratio.

Eynstone

Just use the formula for nth fibonacci number.

billy2908

It's not that simple. I tried fracturing it many times but every time it just seem to leave to the same result.

Matt Benesi

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_n² 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