# 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: **(x ^{2} – x – 1 = 0)** Below are the relation that can be deduced:

=> x

^{2}– x – 1 = 0

=> x^{2}= x + 1

=> x^{3}= x*x^{2}= x*(x+1) = x^{2}+ x = 2x + 1

=> x^{4}= x*x^{3}= x*(2x+1) = 2x^{2}+ x = 2(x+1) + x = 3x + 2

=> x^{5}= x*x^{4}= x*(3x+2) = 3x^{2}+ 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**and

^{(N – 1)}**x**. The same pattern can be observed in the remaining term also. So, the next power of

^{(N – 2)}**x**can be directly expressed as:

=> x = x

=> x^{2}= x+1

=> x^{3}= 2x + 1

=> x^{4}= 3x + 2

=> x^{5}= 5x + 3

=> x^{6}= 8x + 5

=> x^{7}= 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_{N}x + f_{(N – 1)}

where, f_{N}is the nth term in the Fibonacci sequence (n > 0).

Now, Let the roots of the equation: **(x ^{2} – x – 1 = 0)** are ∝ and β, then

∝ = (1 + √5)/2

β = (1 – √5)/2

It can be said that:

=> ∝

^{2}– ∝ – 1 = 0 and β^{2}– β – 1 = 0

=> ∝^{n}= f_{n}∝ + f_{n-1}and β^{n}= f_{n}β + f_{n-1}

=> ∝^{n}– β^{n}= f_{n}(∝ – β)

=> f_{n}= (∝^{n}– β^{n}) / (∝ – 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 Nautilos shell, sunflower, etc.**Binet’s formula:**It is used to find the**N**term in the Fibonacci sequence, which makes it really useful in Mathematics and many fields of computer science as well.^{th}**Golden Ratio**and**Binet’s formula:**They are also used in calculating the time complexities of algorithms like the Euclidean Algorithm etc.