{{Short description|Mathematical sequence of numbers}}
[[File:Geometric sequences.svg|thumb|right|300px|Diagram illustrating three basic geometric sequences of the pattern 1(''r''<sup>''n''−1</sup>) up to 6 iterations deep. The first block is a unit block and the dashed line represents the [[#Series|infinite sum]] of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.]]

A '''geometric progression''', also known as a '''geometric sequence''', is a [[mathematics|mathematical]] [[sequence]] of non-zero [[number]]s where each term after the first is found by multiplying the previous one by a fixed number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2.

Examples of a geometric sequence are [[exponentiation|powers]] ''r''<sup>''k''</sup> of a fixed non-zero number ''r'', such as [[power of two|2<sup>''k''</sup>]] and 3<sup>''k''</sup>. The general form of a geometric sequence is

:<math>a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots</math>

where ''r'' is the common ratio and ''a'' is the initial value.

The sum of a geometric progression's terms is called a ''[[geometric series]]''. Because each two successive numbers in the progression have the same [[Proportionality (mathematics)|proportion]], the terms in a geometric series are also said to be in '''continued proportion''', especially in the context of [[ancient Greek mathematics]], where geometric progressions were described in this way rather than through powers of the common ratio.{{r|artmann}}

==Properties==
The ''n''th term of a geometric sequence with initial value ''a'' = ''a''<sub>1</sub> and common ratio ''r'' is given by
<math display="block"> a_n = a\,r^{n-1},</math>
and in general
<math display="block"> a_n = a_m\,r^{n-m}.</math>
Geometric sequences satisfy the linear [[recurrence relation]]
<math display="block"> a_n = r\,a_{n-1}</math> for every integer <math>n > 1.</math>
This is a first-order, homogeneous [[linear recurrence with constant coefficients]].

Geometric sequences also satisfy the nonlinear recurrence relation
<math display="block"> a_{n} = a_{n-1}^2 / a_{n-2} </math>
for every integer <math>n > 2</math>. This is a second-order nonlinear recurrence with constant coefficients.

When the common ratio of a geometric sequence is positive, the sequence's terms will all share the sign of the first term. When the common ratio of a geometric sequence is negative, the sequence's terms alternate between positive and negative; this is called an alternating sequence. For instance, the sequence 1, −3, 9, −27, 81, −243, ... is an alternating geometric sequence with an initial value of 1 and a common ratio of −3. When the initial term and common ratio are complex numbers, the terms' [[complex argument]]s follow an [[arithmetic progression]].

If the [[absolute value]] of the common ratio is smaller than 1, the terms will decrease in magnitude and approach zero via an [[exponential decay]]. If the absolute value of the common ratio is greater than 1, the terms will increase in magnitude and approach [[infinity]] via an [[exponential growth]]. If the absolute value of the common ratio equals 1, the terms will stay the same size indefinitely, though their signs or complex arguments may change.

Geometric progressions show exponential growth or exponential decline, as opposed to arithmetic progressions showing [[linear function (calculus)|linear]] growth or linear decline. This comparison was taken by [[Thomas Robert Malthus|T.R. Malthus]] as the mathematical foundation of his ''[[An Essay on the Principle of Population]]''. The two kinds of progression are related through the [[exponential function]] and the [[logarithm]]: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression yields an arithmetic progression. The relation that the logarithm provides between a geometric progression in its [[argument of a function|argument]] and an [[arithmetic progression]] of values, prompted [[A. A. de Sarasa]] to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in [[prosthaphaeresis]], leading to the term "hyperbolic logarithm", a synonym for natural logarithm.

== Geometric series ==
{{main article|Geometric series}}
[[File:GeometricSquares.svg|thumb|upright=1|An illustration of an infinite geometric series <math display="inline"> \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + \cdots </math>. It is convergent.]]
A [[geometric series]] is a [[Series (mathematics)|series]] derived from the number of terms <math> a + ar + ar^2, \dots + ar^n </math>.{{r|young}} The geometric series can be either finite or infinite. The infinite geometric series sums the infinitely many terms of a geometric progression. The calculation can either have a result ([[Convergent series|convergence]]) or otherwise ([[Divergent geometric series|divergence]]). An infinite geometric series is convergent if the absolute value of a common ratio is less than one <math> |r| < 1 </math>, showing that the terms approach zero and the partial sums become smaller in magnitude, Likewise, an infinite geometric series is divergent if the absolute value of a common ratio is greater than one <math> |r| > 1 </math>, indicating the terms and the partial sums become bigger in magnitude.{{r|vpr}} An infinite geometric series when <math> |r| = 1 </math> shows its terms become no larger or smaller in magnitude, and the sequence of partial sums of the series does not converge.

==Product==
The infinite product of a geometric progression is the product of all of its terms. The partial product of a geometric progression up to the term with power <math>n</math> is

<math display="block">\prod_{k=0}^{n} ar^{(k)} = a^{n + 1} r^{n(n+1)/2}.</math>

When <math>a</math> and <math>r</math> are positive real numbers, this is equivalent to taking the [[geometric mean]] of the partial progression's first and last individual terms and then raising that mean to the power given by the number of terms <math>n + 1.</math>

<math display="block">\prod_{k=0}^{n} ar^k = a^{n+1} r^{n(n+1)/2} = (\sqrt{a^{2}r^n})^{n+1} \text{ for } a \geq 0, r \geq 0.</math>

This corresponds to a similar property of sums of terms of a finite [[arithmetic sequence]]: the sum of an arithmetic sequence is the number of terms times the [[arithmetic mean]] of the first and last individual terms. This correspondence follows the usual pattern that any arithmetic sequence is a sequence of logarithms of terms of a geometric sequence and any geometric sequence is a sequence of exponentiations of terms of an arithmetic sequence. Sums of logarithms correspond to products of exponentiated values.

===Proof===

Let <math>P_n</math> represent the product up to power <math>n</math>. Written out in full,

:<math>P_n = a \cdot ar \cdot ar^2 \cdots ar^{n-1} \cdot ar^n</math>.

Carrying out the multiplications and gathering like terms,

:<math>P_n = a^{n+1} r^{1+2+3+ \cdots +(n-1)+n}</math>.

The exponent of {{mvar|r}} is the sum of an arithmetic sequence. Substituting the formula for that sum,

:<math>P_n = a^{n+1} r^\frac{n(n+1)}{2}</math>,

which concludes the proof.

One can rearrange this expression to

:<math>P_n = (ar^\frac{n}{2})^{n+1}.</math>

Rewriting {{mvar|a}} as <math>\textstyle \sqrt{a^2}</math> and {{mvar|r}} as <math>\textstyle \sqrt{r^2}</math> though this is not valid for <math>a < 0</math> or <math>r < 0,</math>

:<math>P_n = (\sqrt{a^{2}r^n})^{n+1} \text{ for } a \geq 0, r \geq 0</math>

which is the formula in terms of the geometric mean.

==History==
A clay tablet from the [[Early Dynastic Period (Mesopotamia)|Early Dynastic Period in Mesopotamia]] (c. 2900 – c. 2350 BC), identified as MS 3047, contains a geometric progression with base 3 and multiplier 1/2. It has been suggested to be [[Sumer]]ian, from the city of [[Shuruppak]]. It is the only known record of a geometric progression from before the time of old [[Babylonian mathematics#Old Babylonian mathematics (2000–1600 BC)|Babylonian mathematics]] beginning in 2000 BC.{{r|friberg}}

Books VIII and IX of [[Euclid]]'s [[Euclid's Elements|''Elements'']] analyze geometric progressions (such as the [[powers of two]], see the article for details) and give several of their properties.{{r|heath|artmann}}

== See also ==

* {{annotated link|Arithmetic progression}}
* {{annotated link|Arithmetico-geometric sequence}}
* {{annotated link|Linear difference equation}}
* {{annotated link|Exponential function}}
* {{annotated link|Harmonic progression (mathematics)|Harmonic progression}}
* {{annotated link|Harmonic series (mathematics)|Harmonic series}}
* {{annotated link|Series (mathematics)|Infinite series}}
* {{annotated link|Preferred number}}
* {{annotated link|Thomas Robert Malthus}}
* {{annotated link|Geometric distribution}}

==References==
<references>

<ref name="artmann">{{cite book
 | last = Artmann | first = Benno
 | year = 2012 | orig-year = 1999
 | title = Euclid: The Creation of Mathematics
 | publisher = Springer
 | location = New York
 | isbn = 978-1-4612-1412-0
 | doi = 10.1007/978-1-4612-1412-0_19
 | chapter = Euclid Book VIII: Numbers in Continued Proportion, the Geometry of Numbers
}}</ref>

<ref name="friberg">{{cite book
 | last = Friberg | first = Jöran
 | editor1-last = Friberg | editor1-first = Jöran
 | contribution = MS 3047: An Old Sumerian Metro-Mathematical Table Text
 | doi = 10.1007/978-0-387-48977-3
 | isbn = 978-0-387-34543-7
 | mr = 2333050
 | pages = 150–153
 | publisher = Springer
 | location = New York
 | series = Sources and Studies in the History of Mathematics and Physical Sciences
 | title = A remarkable collection of Babylonian mathematical texts
 | year = 2007
}}</ref>

<ref name="heath">{{cite book
 | last = Heath | first = Thomas L. | author-link = T. L. Heath
 | title = The Thirteen Books of Euclid's Elements
 | url = https://archive.org/details/thirteenbooksofe00eucl
 | url-access = registration
 | edition = 2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925]
 | year = 1956
 | publisher = Dover Publications
 | location = New York
}}</ref>

<ref name="vpr">{{cite book
 | last1 = Varberg | first1 = Dale E.
 | last2 = Purcell | first2 = Edwin J.
 | last3 = Rigdon | first3 = Steven E.
 | title = Calculus
 | year = 2007
 | publisher = [[Pearson Prentice Hall]]
 | edition = 9th
 | page = 456
 | url = https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/456
 | isbn = 978-0131469686
}}</ref>

<ref name="young">{{cite book
 | last = Young | first = Cynthia Y.
 | year = 2010
 | title = Precalculus
 | isbn = 978-0-470-55665-8
 | publisher = [[John Wiley & Sons]]
 | url = https://books.google.com/books?id=9HRLAn326zEC&pg=PA966
 | page = 966
}}</ref>

</references>

*Hall & Knight, ''Higher Algebra'', p.&nbsp;39, {{ISBN|81-8116-000-2}}

==External links==
*{{springer|title=Geometric progression|id=p/g044290}}
*{{mathworld|urlname=GeometricSeries|title=Geometric Series}}

{{Series (mathematics)}}

[[Category:Sequences and series]]
[[Category:Series (mathematics)]]
[[Category:Articles containing proofs]]