# The Fibonacci Sequence

The Fibonacci sequence behaves in the following manner:

Everything starts out very simply.

We begin with two numbers, 0 and 1.

In order to create this sequence of numbers, we **add the last 2 numbers we had**, 0 & 1, and we add them together:

0 + 1 = 1

And our sequence now looks as follows:

[0, 1, 1]

In a more mathematical language, we describe the fibonacci sequence by defining a function **f(n) **in the following way:

**f(0) = 0**

**f(1) = 1**

and for all values where **n** is greater than 2, **n>2**:

**f(n) = f(n-1) + f(n-2)**

This mathematically says that** each number in the sequence is the sum of the previous 2.**

We continue this pattern, adding the last two numbers in the sequence together to get the next number for our sequence:

1 + 1 = 2 [0, 1, 1, 2] **n=3**

and we repeat:

1 + 2 = 3 [0, 1, 1, 2, 3] **n=4**

2 + 3 = 5 [0, 1, 1, 2, 3, 5] **n=5**

3 + 5 = 8 [0, 1, 1, 2, 3, 5, 8] **n=6**

5 + 8 = 13 [0, 1, 1, 2, 3, 5, 8, 13] **n=7**

8 + 13 = 21 [0, 1, 1, 2, 3, 5, 8, 13, 21] **n=8**

13 + 21 = 34 [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] **n=9**

And thus, the sequence expands, adding a possibly infinite number of terms, each the sum of the prior two.

[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...]

From this infinitely-expanding series, we next derive the Golden Ratio.