Fibonacci’s Sequence

A Natural Occurrence

by

Brooke Norman

**What
is the Fibonacci sequence?**

Fibonacci
is a nickname for Leonardo Pisano.
He is the man who discovered the pattern of adding two previous numbers together
will result in the next number of the pattern, starting at 0 and then 1. The general formula for this
sequence is f_{n+1 }= f_{n} + f_{n-1, }f_{1} = f_{2}
=1. The resulting sequence looks
like the following. 1, 1, 2, 3, 5,
13, 21, 34, 55, 89, 144, …

Here is an example of the first 25 numbers:

(this was done on Microsoft Excel)

0 |
0 |

1 |
1 |

2 |
1 |

3 |
2 |

4 |
3 |

5 |
5 |

6 |
8 |

7 |
13 |

8 |
21 |

9 |
34 |

10 |
55 |

11 |
89 |

12 |
144 |

13 |
233 |

14 |
377 |

15 |
610 |

16 |
987 |

17 |
1597 |

18 |
2584 |

19 |
4181 |

20 |
6765 |

21 |
10946 |

22 |
17711 |

23 |
28657 |

24 |
46368 |

Another part of the Fibonacci sequence is the
spiral that it makes. This is made
by using the golden ratio. To
better see the relationship the sequence has in a more geometrical way, we can
look at it as squares. If we start with a square of sides 1, we add another
square of sides 1. The result is two.
This is the new square with sides 2. If the squares of sides 1 and 2 are added, we get the square
of sides 3. This is continued for
adding the square of sides 2 and the square of sides 3 to get a square of sides
5. This continues on and we
continue to get a spiral.

Here
is an example.

What
many people don’t know is that this sequence is found in nature. It occurs quite often and is sometimes
referred to as a "law of nature."

Click
the following links to see different examples in nature.

·
Flowers

Fibonacci
would illustrate his sequence by asking people to solve his “Rabbit Problem.”

How
many pairs of rabbits will be produced in a year, beginning with a single pair,
if in every month each pair bears a new pair which becomes productive from the
second month on?

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