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Geometric progression

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A sequence of numbers each one of which is equal to the preceding one multiplied by a number (the denominator of the progression). A geometric progression is called increasing if q>1, and decreasing if 0<q<1; if q<0, one has a sign-alternating progression. Any term of a geometric progression a_j can be expressed by its first term a_0 and the denominator q by the formula \begin{equation} a_j=a_0q^{j}. \end{equation} The sum of the first n terms of a geometric progression (with q\ne1) is given by the formula \begin{equation} a_0+a_0q+a_0q^2+\dots+a_0q^{n-1}= S_n = a_0\frac{1-q^n}{1-q}= \frac{a_n-a_0}{q-1} \end{equation} If |q|<1, the sum S_n tends to the limit S=a_0/(1-q) as n tends to infinity. This number S is known as the sum of the infinitely-decreasing geometric progression.

The expression \begin{equation} a_0+a_0q+a_0q^2+\dots+a_0q^{n}+\dots, \end{equation} if |q|<1 is the simplest example of a convergent series — a geometric series; the number a_0/(1-q) is the sum of the geometric series.

The term "geometric progression" is connected with the following property of any term of a geometric progression with positive terms: a_n = \sqrt{a_{n-1}a_{n+1}}, i.e. any term is the geometric mean of the term which precedes it and the term which follows it.

How to Cite This Entry:
Geometric progression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometric_progression&oldid=29220
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article