## Explaining the Fibonacci Sequence

The Fibonacci Sequence is the sequence of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811

Terms of the sequence are found by adding the preceding two numbers, so that the 4th and 5th terms are added to give the 6th term, 5.

When you make squares with sides equal to terms in the Fibonacci sequence, we can trace a spiral through the corners of the squares as shown below.

If you divide successive Fibonacci Numbers, the bigger being divided by the smaller, their ratio is very close to the Golden Ratio which is approximately

The bigger the pair of Fibonacci Numbers, the better the approximation.

A | B | B / A |
---|---|---|

2 | 3 | 1.5 |

3 | 5 | 1.666666666... |

5 | 8 | 1.6 |

8 | 13 | 1.625 |

... | ... | ... |

144 | 233 | 1.618055556... |

233 | 377 | 1.618025751... |

... | ... | ... |