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Geometric progression

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Mathematical sequence of numbers

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 theinfinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.

Ageometric progression, also known as ageometric sequence, is amathematical sequence of non-zeronumbers where each term after the first is found by multiplying the previous one by a fixed number called thecommon 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 arepowersr^k of a fixed non-zero numberr, such as2^k and 3^k. The general form of a geometric sequence is

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

wherer is the common ratio anda is the initial value.

The sum of a geometric progression's terms is called ageometric series.

Properties

Thenth term of a geometric sequence with initial valuea =a₁ and common ratior is given by

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

and in general

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

Geometric sequences satisfy the linearrecurrence relation

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

for every integer

n>1.

This is a first order, homogeneouslinear 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 anarithmetic progression.

If theabsolute value of the common ratio is smaller than 1, the terms will decrease in magnitude and approach zero via anexponential decay. If the absolute value of the common ratio is greater than 1, the terms will increase in magnitude and approachinfinity via anexponential 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 showinglinear growth or linear decline. This comparison was taken byT.R. Malthus as the mathematical foundation of hisAn Essay on the Principle of Population. The two kinds of progression are related through theexponential function and thelogarithm: 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 ageometric progression in itsargument and anarithmetic progression of values, promptedA. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms inprosthaphaeresis, leading to the term "hyperbolic logarithm", a synonym for natural logarithm.

Geometric series

Proof without words of the formula for the sum of a geometric series –if |r| < 1 andn → ∞, therⁿ term vanishes, leavingS_∞ =⁠a/1 −r⁠

This section is an excerpt fromGeometric series.[edit]

Part of a series of articles about

\int _{a}^{b}f'(t)\,dt=f(b)-f(a)

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Inmathematics, ageometric series is aseries summing the terms of an infinitegeometric 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 $1 {\displaystyle 1}$ ⁠. Each term in a geometric series is thegeometric mean of the term before it and the term after it, in the same way that each term of anarithmetic series is thearithmetic mean of its neighbors.

WhileGreek 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 byGreek mathematicians, for example used byArchimedes tocalculate the area inside a parabola (3rd century BCE). Today, geometric series are used inmathematical finance, calculating areas of fractals, and various computer science topics.

Though geometric series most commonly involvereal orcomplex numbers, there are also important results and applications formatrix-valued geometric series, function-valued geometric series,

p {\displaystyle p}

-adic number geometric series, and most generally geometric series of elements of abstract algebraicfields,rings, andsemirings.

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 $n {\displaystyle n}$ is

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

When $a {\displaystyle a}$ and $r {\displaystyle r}$ are positive real numbers, this is equivalent to taking thegeometric 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 finitearithmetic sequence: the sum of an arithmetic sequence is the number of terms times thearithmetic 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 $n {\displaystyle n}$ . 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 ofr 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}.

Rewritinga as $\textstyle {\sqrt {a^{2}}}$ andr 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 theEarly 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 beSumerian, from the city ofShuruppak. It is the only known record of a geometric progression from before the time of oldBabylonian mathematics beginning in 2000 BC.^[1]

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

See also

Arithmetic progression – Sequence of equally spaced numbers
Arithmetico-geometric sequence – Mathematical sequence satisfying a specific pattern
Linear difference equation – Relation in AlgebraPages displaying short descriptions of redirect targets
Exponential function – Mathematical function, denoted exp(x) or e^x
Harmonic progression – Progression formed by taking the reciprocals of an arithmetic progression
Harmonic series – Divergent sum of all positive unit fractions
Infinite series – Infinite sum
Preferred number – Standard guidelines for choosing exact product dimensions within a given set of constraints
Thomas Robert Malthus – British political economist (1766–1834)
Geometric distribution – Probability distribution

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

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Sequences andseries

Integer sequences

Basic	Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced(list)	Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

Fibonacci spiral with square sizes up to 34.

Properties of sequences

Properties of series

Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2^s + 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric series

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