Why the value of Golden Ratio is 1.618 and how is it related to Binet’s formula ?

Golden Ratio: Two numbers, say A and B are said to be in the golden ratio if their ratio equals the ratio of the sum of two numbers to the larger number, i.e.,

Suppose A > B, then
If A/B = (A + B)/A = ∅ = 1.618(Golden Ratio), 
then these two numbers are said to be in golden ratio. 

It is denoted by ∅ and its value is equal to 1.6180339…, which is an Irrational Number.

Binet’s Formula: This formula is used to find the N^th term in the Fibonacci Sequence which is given by:

where, F_N is the N^th term in the Fibonacci Sequence.

For the equation: (x² – x – 1 = 0) Below are the relation that can be deduced:

=> x² – x – 1 = 0
=> x² = x + 1
=> x³ = x*x² = x*(x+1) = x² + x = 2x + 1
=> x⁴ = x*x³ = x*(2x+1) = 2x² + x = 2(x+1) + x = 3x + 2
=> x⁵ = x*x⁴ = x*(3x+2) = 3x² + 2x = 3(x+1) + 2x = 5x + 3

The next term for the next power of x can be guessed by looking at the above pattern. Observe that the coefficient of x^N is equal to the sum of the coefficient of x^{(N – 1)} and x^{(N – 2)}. The same pattern can be observed in the remaining term also. So, the next power of x can be directly expressed as:

=> x = x
=> x² = x+1
=> x³ = 2x + 1
=> x⁴ = 3x + 2
=> x⁵ = 5x + 3
=> x⁶ = 8x + 5
=> x⁷ = 13x + 8
…

The Fibonacci Sequence is given by {0, 1, 1, 2, 3, 5, 8, 13, 21, …, }, and there exists a relation between the two, after observing the above two sequences. It can be said that:

x^N = f_Nx + f_{(N – 1)}
where, f_N is the nth term in the Fibonacci sequence (n > 0).

Now, Let the roots of the equation: (x² – x – 1 = 0) are ∝ and β, then

∝ = (1 + √5)/2
β = (1 – √5)/2

It can be said that:

=> ∝² – ∝ – 1 = 0 and β² – β – 1 = 0
=> ∝ⁿ = f_n∝ + f_n-1 and βⁿ = f_nβ + f_n-1
=> ∝ⁿ – βⁿ = f_n(∝ – β)
=> f_n = (∝ⁿ – βⁿ) / (∝ – B)

After substituting the values of ∝ and β in the above equation:

The above equation is known as Binet’s Formula. And the value (1+√5)/2 is known as the Golden Ratio, which is equal to 1.618. Therefore, the N^th Fibonacci Number is given by:

F_N ≈ ∅^N
where, where, ∅ is the Golden Ratio and F_n is the nth Fibonacci term.

Applications:

• Golden Ratio: It is used in architecture, paintings, photography and is also present in many forms in nature itself like in the Nautilus shell, sunflower, etc.
• Binet’s formula: It is used to find the N^th term in the Fibonacci sequence, which makes it really useful in Mathematics and many fields of computer science as well.
• Golden Ratio and Binet’s formula: They are also used in calculating the time complexities of algorithms like the Euclidean Algorithm etc.