# Formulas and properties of geometric sequence.

Definition

**Geometric sequence**(

**geometric progression**) — a sequence of numbers b

_{1}, b

_{2}, b

_{3}, ..., in which each member, starting with the second, equal to the product of the previous member and a constant number q (

*common ratio*), where b

_{1}≠ 0, q ≠ 0.

## n -th term of the geometrical sequence is given by

b_{n}= b

_{1}· q

^{n - 1}

b

_{n}= b

_{n - 1}· q

## Common ratio

q = | b_{n} |

b_{n - 1} |

A

**geometric series**is the*sum*of the numbers in a geometric sequence.## Formulas of geometric series

S_{n} = |
b_{1} - b_{n + 1} |

1 - q |

S_{n} = b_{1} · |
1 - q^{n} |

1 - q |

## Properties of geometric sequence

b_{n}

^{2}= b

_{n + 1}· b

_{n - 1}

## An infinite geometric series

If |q| < 1 and n → ∞S = | b_{1} |

1 - q |

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