Question

The ratio as n gets larger is aid to approach the Golden ratio, which is approximately equal to 1.618. What happens to the inverse of this ratio, ? What number does this quantity approach? How does this compare to the original ratio?​

Answer 1

Answer:

The limit of the ratios of the consecutive Fibonacci terms as  n  tends to infinity is the golden number  φ . The inverse of this ratio is  1/φ  which equals to  φ−1 .

Proof:

We will employ Binet’s formula from which we know that:

φ=(5–√+1)/2…(1)  

ψ=(1−5–√)/2…(2)  

Lim[Fn+1/Fn]=  

[(φn+1−ψn+1)/5–√]/[(φn−ψn)/5–√]=  

Lim[(φn+1−ψn+1)]/[(φn−ψn)]…(3)  

We divide numerator and denominator by  φn  and we take:

Lim[(φn+1−ψn+1)]/[(φn−ψn)]=  

Lim[(φ−[(1−5–√)/(1+5–√)]n)((1−5–√)/2)]/[1−[(1−5–√)/(1+5–√)]n]=  

Lim[(φ−0)/(1–0)]=φ  

Proof that  1/φ=φ−1 :

1/φ=1/[(5–√+1)/2]=2/(5–√+1)=  

2(5–√−1)/[(5–√+1)(5–√−1)]=  

2(5–√−1)/(5−1)=(5–√−1)/2=  

(5–√+1−2)/2=(5–√+1)/2−2/2=φ−1  

The same happens with the Lucas sequence and with any Generalized Fibonacci sequence.

Step-by-step explanation: