# values of $\frac{{F}_{n}}{{F}_{n-1}}$ for $$

The golden ratio^{} $\varphi $ is an irrational number, approximately 1.6180339887498948482045868. Dividing one Fibonacci number^{} by the previous gives better and better approximations to the golden ratio the bigger the numbers get. One division gets a little under, then the next a little over, as the table below shows. 144 divided by 89 is probably good enough for most practical purposes.

${F}_{n}$ | ${F}_{n-1}$ | $\frac{{F}_{n}}{{F}_{n-1}}$ to 20 decimal places | Approximation error |
---|---|---|---|

1 | 1 | 1.00000000000000000000 | -0.61803398874989484820 |

2 | 1 | 2.00000000000000000000 | 0.38196601125010515179 |

3 | 2 | 1.50000000000000000000 | -0.11803398874989484820 |

5 | 3 | 1.66666666666666666666 | 0.04863267791677181846 |

8 | 5 | 1.60000000000000000000 | -0.01803398874989484820 |

13 | 8 | 1.62500000000000000000 | 0.00696601125010515179 |

21 | 13 | 1.61538461538461538461 | -0.00264937336527946358 |

34 | 21 | 1.61904761904761904761 | 0.00101363029772419941 |

55 | 34 | 1.61764705882352941176 | -0.00038692992636543643 |

89 | 55 | 1.61818181818181818181 | 0.00014782943192333361 |

144 | 89 | 1.61797752808988764044 | -0.00005646066000720775 |

233 | 144 | 1.61805555555555555555 | 0.00002156680566070735 |

377 | 233 | 1.61802575107296137339 | -0.00000823767693347481 |

610 | 377 | 1.61803713527851458885 | 0.00000314652861974065 |

987 | 610 | 1.61803278688524590163 | -1.20186464894656524257 |

1597 | 987 | 1.61803444782168186423 | 0.00000045907178701603 |

2584 | 1597 | 1.61803381340012523481 | -0.00000017534976961338 |

4181 | 2584 | 1.61803405572755417956 | 0.00000066977659331361 |

6765 | 4181 | 1.61803396316670652953 | -0.00000002558318831866 |

10946 | 6765 | 1.61803399852180339985 | 0.00000000977190855164 |

17711 | 10946 | 1.61803398501735793897 | -0.00000000373253690923 |

28657 | 17711 | 1.61803399017559708655 | 0.00000000142570223835 |

46368 | 28657 | 1.61803398820532505147 | -0.00000000054456979673 |

75025 | 46368 | 1.61803398895790200138 | 0.00000000020800715317 |

Title | values of $\frac{{F}_{n}}{{F}_{n-1}}$ for $$ |
---|---|

Canonical name | ValuesOffracFnFn1For1N26 |

Date of creation | 2013-03-22 17:23:54 |

Last modified on | 2013-03-22 17:23:54 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Example |

Classification | msc 40A05 |

Classification | msc 11B39 |