# Fibonacci and golden section proof

• Jan 11th 2010, 02:02 PM
ozfingwe
Fibonacci and golden section proof
hello,

based on the formular for Fibonacci numbers I am supposed to calculate the value for the golden section phi.
(1) rn = Fn+1 / Fn converges against phi, and I have to prove that
(2) rn+1 = 1 + 1 / rn and thus phi = 1 + 1 / phi

But how do I get from (1) to (2) ?

any help greatly appreciated!

oz.
• Jan 11th 2010, 02:09 PM
Drexel28
Quote:

Originally Posted by ozfingwe
hello,

based on the formular for Fibonacci numbers I am supposed to calculate the value for the golden section phi.
(1) rn = Fn+1 / Fn converges against phi, and I have to prove that
(2) rn+1 = 1 + 1 / rn and thus phi = 1 + 1 / phi

But how do I get from (1) to (2) ?

any help greatly appreciated!

oz.

There is a "better" way to do this, if you are curious. But just note that $r_{n+1}=\frac{F_{n+2}}{F_{n+1}}=\frac{F_{n+1}+F_{n }}{F_{n+1}}=1+\frac{F_n}{F_{n+1}}=1+\frac{1}{r_n}$
• Jan 11th 2010, 02:12 PM
ozfingwe
just what I had in mind and couldn't put onto paper, thanks a lot mate!
oz.