Approximation to Golden Ratio

We know that the golden ratio is $$\alpha : = {{1 + \sqrt 5 } \over 2} = 1.618...$$ and we claim that $$\mathop {\lim }\limits_{n \to \infty } {{{F_{n + 1}}} \over {{F_n}}} = \alpha$$ where F represents the Fibonacci Sequences

First of all,let us define the Fibonacci Sequences.

$${F_n} = {F_{n - 2}} + {F_{n - 1}},n \ge 2,{F_0} = 0,{F_1} = 1$$

First of all, define the Fibonacci Sequences in Python
            
f_l=[0,1]
ht=int(input('Enter n:'))
for i in range(ht):
    f_l.append(f_l[i]+f_l[i+1])
    print(f_l[i+2]/f_l[i+1])#Ratio
Let us consider first 17 term of the sequence(n=15)
print(f_l)#Output will be [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987]

For n=1 and n=2 terms of the Fibonacci Sequences, ratio is 1 i.e. $${{{F_2}} \over {{F_1}}} = 1$$ For n=3 and n=4 terms of the Fibonacci Sequences, ratio is 1.5 i.e. $${{{F_3}} \over {{F_4}}} = 1.5$$ . . . For n=13 and n=14 terms of the Fibonacci Sequences, ratio is 1.618 i.e. $${{{F_13}} \over {{F_14}}} = 1.618...$$ For n=15 and n=16 terms of the Fibonacci Sequences, ratio is 1.618 i.e. $${{{F_15}} \over {{F_16}}} = 1.618..$$ This behaviour goes through the same way. Calculations of ratio with Python was shown in the above source code.

An Approximation to Golden Ratio With Python