Connection Between Fibonacci Numbers and The Golden Ratio

Welcome.

In this blog I will tell you relation between the Fibonacci Sequences and the Golden Ratio.Before that below  you can see the statue of Fibonacci, made in 1863 by Giovanni Paganucci, a sculpture in florence, but kept in a ancient cemetery in pisa where the Fibonacci was born. Its is interesting that the likeness of Fibonacci in this statue and his iconic portrait probably looks nothing like Fibonacci, since no true drawings of him exist from 850 years ago .

But nevertheless Italy still honors him with this sculpture. 

Statue Of Fibonacci

Let's Return to Fibonacci Numbers and Fibonacci recursion relation. Then we will show you how they are related to Golden Ratio. So let's do some mathematics.

Any way what's the recursion relation do you remember??

The n+1 Fibonacci number is equal to the sum of preceding two that is nth Fibonacci number Plus n-1 Fibonacci number right .

                       Fn+1 = Fn + Fn−1   ........(1)

To show the relationship between Fibonacci number and the Golden Ratio, we need to look at the ratios for that divide equation 1 by Fn. We get ,

         

     ...(2)

This equation by itself is difficult to solve. But we can look at the limit of this equation when n becomes larger that is when n tends to infinity. 

So we need to make an assumption here. We say the limit as n goes to infinity, n is very large, of the ratios of two consecutive Fibonacci numbers, Fn+1 divide by Fn we say that make the assumption that this limit exists we can call the limit alpha α . With this assumption let's solve equation 2 when n goes to infinity we get ,

       

Applying the limits we get 

           

  

                         α = 1 + 1/α          .... (3)

We know left hand side is equal to α, where as right hand side limit is smaller number over larger one so its equal to 1 over α. Because we can write left side limit as 1 over larger number over smaller than Apply limit we get 1 over alpha. 

Here equation 3 is the same equation we derived for Golden Ratio. Capital phi equal to 1 plus 1 over capital phi, right ?? Where 1 over capital phi is Golden  Ratio Conjugate. 

So this Ratio is equal to Golden Ratio.

So the limit of two consecutive Fibonacci numbers, when n tends to infinity is equal to the Golden Ratio.

That's the important relation between Fibonacci number and Golden Ratio. Let's look at some Numerical values.

Here we list n from 1 to 10, and we find Ratio of  n+1 Fibonacci number to nth Fibonacci number. 

        

Ratio Of Two Consecutive Fibonacci Numbers approaches to Golden Ratio 

Here we are getting Convergence , very slow convergence to Golden Ratio. We remember the Golden Ratio approx 1.618 something. Here Convergence is going positive , negative, positive,  negative on both side of Golden Ratio. In the last column we can see the difference between Ratio of two consecutive Fibonacci numbers and The Golden Ratio, its negative , positive, negative,  positive continuing and this number are getting closer and closer to Zero.

So the Ratios of two Consecutive Fibonacci numbers eventually converging or tends to Golden Ratio . That's the clear connection between them.

We can derive a formula for the Fibonacci numbers that contains the Golden Ratio. This formula is known as Binet's formula.

We will cover it in our next Blog.

Happy Learning Thanks for reading.