# 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.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the **DSA Self Paced Course** at a student-friendly price and become industry ready. To complete your preparation from learning a language to DS Algo and many more, please refer **Complete Interview Preparation Course****.**

In case you wish to attend **live classes **with experts, please refer **DSA Live Classes for Working Professionals **and **Competitive Programming Live for Students**.