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

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  • Last Updated : 19 Apr, 2021

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 Nth term in the Fibonacci Sequence which is given by:

F_N = \frac{({\frac{1 + \sqrt 5}{2}})^N + ({\frac{1 - \sqrt 5}{2}})^N}{2}

where, FN is the Nth 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 xN 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:

xN = fNx + f(N – 1)
where, fN 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 = fn∝ + fn-1 and βn = fnβ + fn-1
=> ∝n – βn = fn(∝ – β)
=> fn = (∝n – βn) / (∝ – B)

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

F_N = \frac{({\frac{1 + \sqrt 5}{2}})^N + ({\frac{1 - \sqrt 5}{2}})^N}{2}

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 Nth Fibonacci Number is given by:

FN ≈ ∅N
where, where, ∅ is the Golden Ratio and Fn 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 Nautilos shell, sunflower, etc.
  • Binet’s formula: It is used to find the Nth 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.
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