# geometric sequence

A sequence^{} of the form

$$a,ar,a{r}^{2},a{r}^{3},\mathrm{\dots}$$ |

of real or complex numbers is called geometric sequence^{}. of the geometric sequence is thus that every two consecutive members of the sequence have the constant ratio $r$, called usually the common ratio of the sequence (if $ar=0$, speaking the ratio of members does not exist).

The ${n}^{\mathrm{th}}$ member of the geometric sequence has the

$${a}_{n}=a{r}^{n-1}.$$ |

Let $a\ne 0$. The sequence is convergent^{} for $$ having the limit (http://planetmath.org/LimitOfRealNumberSequence) 0, and for $r=1$ having as constant sequence the limit $a$.

When the members of the sequence are positive numbers, each member is the geometric mean of the preceding and the following member; the name “geometric sequence”(or “geometric series^{}”) is due to this fact (a fact is true for the harmonic series and harmonic mean).

Title | geometric sequence |
---|---|

Canonical name | GeometricSequence |

Date of creation | 2013-03-22 14:38:52 |

Last modified on | 2013-03-22 14:38:52 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 14 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 40-00 |

Related topic | GeometricSeries |

Related topic | LimitOfRealNumberSequence |

Defines | common ratio |