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Difference between revisions of "Geometric progression"

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A sequence of numbers each one of which is equal to the preceding one multiplied by a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442901.png" /> (the denominator of the progression). A geometric progression is called increasing if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442902.png" />, and decreasing if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442903.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442904.png" />, one has a sign-alternating progression. Any term of a geometric progression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442905.png" /> can be expressed by its first term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442906.png" /> and the denominator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442907.png" /> by the formula
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442908.png" /></td> </tr></table>
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A sequence of numbers each one of which is equal to the preceding one multiplied by a number $q\ne0$ (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
 
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\begin{equation}
The sum of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442909.png" /> terms of a geometric progression (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429010.png" />) is given by the formula
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a_j=a_0q^{j}.
 
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\end{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429011.png" /></td> </tr></table>
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The sum of the first $n$ terms of a geometric progression (with $q\ne1$) is given by the formula
 
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\begin{equation}
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429012.png" />, the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429013.png" /> tends to the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429014.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429015.png" /> increases without limit. This number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429016.png" /> is known as the sum of the infinitely-decreasing geometric progression.
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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}
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\end{equation}
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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
 
The expression
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\begin{equation}
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a_0+a_0q+a_0q^2+\dots+a_0q^{n}+\dots,
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\end{equation}
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if $|q|<1$ is the simplest example of a convergent [[Series|series]] — a geometric series; the number $a_0/(1-q)$ is the sum of the geometric series.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429017.png" /></td> </tr></table>
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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|geometric mean]] of the term which precedes it and the term which follows it.
 
 
is the simplest example of a convergent series — a geometric series; the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429018.png" /> 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429019.png" />, i.e. any term is the [[Geometric mean|geometric mean]] of the term which precedes it and the term which follows it.
 

Latest revision as of 14:34, 16 December 2012


A sequence of numbers each one of which is equal to the preceding one multiplied by a number $q\ne0$ (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=12512
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article