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# geometric sequence

A sequence of the form

$a,\,ar,\,ar^{2},\,ar^{3},\,\ldots$ |

of real or complex numbers is called geometric sequence. Characteristic 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 $ar=0$, strictly speaking the ratio of members does not exist).

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

$a_{n}=ar^{{n-1}}.$ |

Let $a\neq 0$. The sequence is convergent for $|r|<1$ having the limit 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 comparable fact is true for the harmonic series and harmonic mean).

## Mathematics Subject Classification

40-00*no label found*

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