## Fibonacci Sequence

Table of Contents

## About Fibonacci (The Man)

His real name was Leonardo Pisano Bogollo. He lived in Italy between 1170 and 1250.

“Fibonacci” was his nickname, which roughly means “Son of Bonacci”.

Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc).

## Definition of Fibonacci Sequence

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

The next number is found by adding up the 2 numbers before it.

Number | Multiplication of 2 Numbers Before |

2 | 1 + 1 |

3 | 1 + 2 |

5 | 2 + 3 |

8 | 3 + 5 |

13 | 5 + 8 |

21 | 8 + 13 |

32 | 13 + 21 |

… | … |

## The Rule of Fibonacci Sequence

Let’s write Fibonacci Sequence a Rule!

First, the terms are numbered from 0 onwards like this:

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

x_{n} | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 |

So term number 7 is called x_{7}, which equals 13.

the 7th term is the 6th term plus the 5th term: x_{7} = x_{6} + x_{5}

So we can write the rule:

**x**_{n}** = x**_{n-1}** + x**_{n-2}

**where:**

- x
_{n}is term number “n” - x
_{n-1}is the previous term (n-1) - x
_{n-2}is the term before that (n-2)

Find the term 10!

term 10 is calculated like this:

x_{10 }= x_{10-1} + x_{10-2}_{}

x_{10 }= x_{9} + x_{8}_{}

x_{10 }= 34 + 21

x_{10 }= 55

## Fibonacci Sequence Terms Below Zero

The sequence works below zero too, like this:

n | … | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | … |

x_{n} | … | -3 | 2 | -1 | 1 | 0 | 1 | 1 | 2 | 3 | … |

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- … pattern. It can be written like this:

**x**_{−n}** = (−1)**^{n+1}** x**_{n}

Which says that term “-n” is equal to (−1)^{n+1} times term “n”, and the value (−1)^{n+1} neatly makes the correct 1,-1,1,-1,… pattern.

## Interesting Pattern of Fibonacci Sequence

Pattern | Elements |

Every 3rd number is a multiple of x_{3} = 2 | 2, 8, 34, 144, 610, … |

Every 4th number is a multiple of x_{4} = 3 | 3, 21, 144, … |

Every 5th number is a multiple of x_{5 }= 5 | 5, 55, 610, … |

… | … |

Every nth number is a multiple of x_{n} |