Fibonacci Sequence - What is it and how is it connected to the golden ratio

As promised, this blog post will be on the Fibonacci Sequence! (If you have yet to read the previous post on the Golden Ratio, do read it first before coming back to this post.)

Why did I post the rabbit photo in the previous post?

If you notice the number sequence on the right of the picture, it goes 1, 1, 2, 3, 5, ...which is actually the Fibonacci Sequence. What this means is that the reproduction cycle of living things, given that they reproduce in pairs, follows the Fibonacci Sequence! That's not all the Fibonacci-linked natural occurrences by the way...

Other Fibonacci-linked natural occurrences

Branching of trees

Spirals in a shell

Curves of a wave

And many many more...

So what exactly is the Fibonacci Sequence?
A sequence that satisfies the following equation is known as a Fibonacci Sequence.

So the sequence is as follows: 

F0 = 0

F1 = 1

F2 = F0 + F1 = 0 +1 = 1

F3 = F1 + F2 = 1 + 1 = 2

F4 = F2 + F3 = 1 + 2 = 3

F5 = F3 + F4 = 2 + 3 = 5

F6 = F4 + F5 = 3 + 5 = 8 

and so on...i.e. each successive number is obtained by adding the sum of the two previous numbers. Easy!

The Fibonacci Sequence Golden Ratio Connection

This is the fascinating part and why Math is fun! Remember the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,... Dividing each number by the previous number gives 1/1 = 1, 2/1 = 2, 3/2 = 1.5, and so on, up to 144/89 = 1.6179. The resulting sequence will then be

1, 2, 1.5 1.6666.., 1.6, ...., 1.6179

You would realise that the numbers go up and down and keeps getting closer to you guessed it, the golden ratio, which is equals to 1.618. 

Because of this connection, mathematicians have actually found a connection that allows you to calculate any Fibonacci Number using the Golden Ratio. I'll let you brood over this equation and end the post here...

where Xn is the n th Fibonacci Number.