Geometric progression

Diagram illustrating three basic geometric sequences of the pattern 1(rⁿ⁻¹) up to 6 iterations deep. The first block is a unit block and the dashed line represents the 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 mathematical sequence of non-zero numbers 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 powers r^k of a fixed non-zero number r, such as 2^k and 3^k. The general form of a geometric sequence is

a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots

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.

Properties

The nth term of a geometric sequence with initial value a = a₁ and common ratio r is given by

a_{n}=a\,r^{n-1},

and in general

a_{n}=a_{m}\,r^{n-m}.

Geometric sequences satisfy the linear recurrence relation

a_{n}=r\,a_{n-1}

for every integer

n>1.

This is a first order, homogeneous linear recurrence with constant coefficients.

Geometric sequences also satisfy the nonlinear recurrence relation

$a_{n}=a_{n-1}^{2}/a_{n-2}$ for every integer $n>2.$

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 arguments 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 growth or linear decline. This comparison was taken by 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.

Geometric series

In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series ${\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{8}}+\cdots$ is a geometric series with common ratio ⁠ ${\tfrac {1}{2}}$ ⁠, which converges to the sum of ⁠ $1$ ⁠. Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors.

While Greek philosopher Zeno's paradoxes about time and motion (5th century BCE) have been interpreted as involving geometric series, such series were formally studied and applied a century or two later by Greek mathematicians, for example used by Archimedes to calculate the area inside a parabola (3rd century BCE). Today, geometric series are used in mathematical finance, calculating areas of fractals, and various computer science topics.

Though geometric series most commonly involve real or complex numbers, there are also important results and applications for matrix-valued geometric series, function-valued geometric series,

p

-adic number geometric series, and most generally geometric series of elements of abstract algebraic fields, rings, and semirings.

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 $n$ is

$\prod _{k=0}^{n}ar^{(k)}=a^{n+1}r^{n(n+1)/2}.$

When $a$ and $r$ 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 $n+1.$

$\prod _{k=0}^{n}ar^{k}=a^{n+1}r^{n(n+1)/2}=({\sqrt {a^{2}r^{n}}})^{n+1}{\text{ for }}a\geq 0,r\geq 0.$

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 $P_{n}$ represent the product up to power $n$ . Written out in full,

P_{n}=a\cdot ar\cdot ar^{2}\cdots ar^{n-1}\cdot ar^{n}

.

Carrying out the multiplications and gathering like terms,

P_{n}=a^{n+1}r^{1+2+3+\cdots +(n-1)+n}

.

The exponent of $r$ is the sum of an arithmetic sequence. Substituting the formula for that sum,

P_{n}=a^{n+1}r^{\frac {n(n+1)}{2}}

,

which concludes the proof.

One can rearrange this expression to

P_{n}=(ar^{\frac {n}{2}})^{n+1}.

Rewriting $a$ as $\textstyle {\sqrt {a^{2}}}$ and $r$ as $\textstyle {\sqrt {r^{2}}}$ though this is not valid for $a<0$ or $r<0,$

P_{n}=({\sqrt {a^{2}r^{n}}})^{n+1}{\text{ for }}a\geq 0,r\geq 0

which is the formula in terms of the geometric mean.

History

A clay tablet from the 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 Sumerian, from the city of Shuruppak. It is the only known record of a geometric progression from before the time of old Babylonian mathematics beginning in 2000 BC.^[1]

Books VIII and IX of Euclid's Elements analyze geometric progressions (such as the powers of two, see the article for details) and give several of their properties.^[2]

References

^ Friberg, Jöran (2007). "MS 3047: An Old Sumerian Metro-Mathematical Table Text". In Friberg, Jöran (ed.). A remarkable collection of Babylonian mathematical texts. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 150–153. doi:10.1007/978-0-387-48977-3. ISBN 978-0-387-34543-7. MR 2333050.
^ Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.

Hall & Knight, Higher Algebra, p. 39, ISBN 81-8116-000-2

External links

"Geometric progression", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Derivation of formulas for sum of finite and infinite geometric progression at Mathalino.com
Geometric Progression Calculator Archived 2008-12-27 at the Wayback Machine
Nice Proof of a Geometric Progression Sum at sputsoft.com
Weisstein, Eric W. "Geometric Series". MathWorld.

[1] Friberg, Jöran (2007). "MS 3047: An Old Sumerian Metro-Mathematical Table Text". In Friberg, Jöran (ed.). A remarkable collection of Babylonian mathematical texts. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 150–153. doi:10.1007/978-0-387-48977-3. ISBN 978-0-387-34543-7. MR 2333050.

[2] Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.

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