Undergrad Fibonacci Series and Golden Ratio Explained

[Kevin McHugh]

I'm not sure this is the right forum, so if not, please move to the appropriate forum. My question is why does the ratio of two consecutive fibonacci numbers converge to the golden ratio? I see no mathematical connection between the series ratios and ratios of a unit line segment divided into two unequal segments. .

 

[mfb]

You can show that
(a) if two positive real numbers have this ratio, then their sum and the larger number have this ratio again
(b) if the ratio is larger, then the ratio of the sum and the larger number is smaller, and vice versa
(c) that the difference to the golden ratio always decreases in the cases of (b)

 

[fresh_42]

The Fibonacci sequence is defined by $F(n+1)=F(n)+F(n-1)$ and therefore
$$ \frac{F(n+1)}{F(n)}= \frac{F(n)+F(n-1)}{F(n)} = 1 + \frac{F(n-1)}{F(n)} $$
and the golden ratio is defined by
$$ \frac{a+b}{a}=\frac {a}{b} =: \varphi $$
If the Fibonacci sequence converges to $\Phi$, then $\Phi = 1 + \frac{1}{\Phi}$.

You may try and find out why $\Phi = \varphi$ or read the following passage:
https://en.wikipedia.org/wiki/Golden_ratio#Relationship_to_Fibonacci_sequence

 

[Kevin McHugh]

Thank you gentlemen. I guess I should have searched the internet first,