Geometric progression

geometric sequencegeometricgeometricalgeometricallyfinite geometric seriesgeometric rategeometric sequencesgeometric seriesgeometric sumgeometrical progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.wikipedia
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Exponential growth

exponentiallyexponentialgrow exponentially
In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression.

Logarithm

logarithmsloglogarithmic function
Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.
The relation that the logarithm provides between a geometric progression in its 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.

Sequence

sequencessequentialinfinite sequence
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Geometric series

infinite geometric seriesGeometric series Formulageometric sum
It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is
The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant.

Summation

sumsumssigma notation
A geometric series is the sum of the numbers in a geometric progression.

Arithmetico–geometric sequence

Arithmetico-geometric sequencearithmetico-geometric series
In mathematics, an arithmetico–geometric sequence is the result of the term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression.

Geometric distribution

geometricgeometrically distributed geometrically distributed
In either case, the sequence of probabilities is a geometric sequence.

Preferred number

1-2-5 seriespreferred numberspreferred values
The factor between two consecutive numbers in a Renard series is approximately constant (before rounding), namely the 5th, 10th, 20th, or 40th root of 10 (approximately 1.58, 1.26, 1.12, and 1.06, respectively), which leads to a geometric sequence.

Euclid's Elements

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

Power of two

powers of twopower of 2powers of 2
Books VIII and IX of Euclid's Elements analyzes geometric progressions (such as the powers of two, see the article for details) and give several of their properties. Examples of a geometric sequence are powers r k of a fixed number r, such as 2 k and 3 k.

Mathematics

mathematicalmathmathematician
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Number

number systemnumericalnumbers
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Exponentiation

exponentpowerpowers
Examples of a geometric sequence are powers r k of a fixed number r, such as 2 k and 3 k.

Scale factor

scalescaledscaling factor
where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.

Recurrence relation

difference equationdifference equationsrecurrence relations
Such a geometric sequence also follows the recursive relation

Linear function (calculus)

linearLinear functionlinear functions
Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11).

Thomas Robert Malthus

Thomas MalthusMalthusRobert Malthus
This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population.

Derivative

differentiationdifferentiablefirst derivative
Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form

Stirling numbers of the second kind

Stirling number of the second kindStirling numbergenerating function of Stirling numbers of the second kind
An exact formula for the generalized sum G_s(n, r) when is expanded by the Stirling numbers of the second kind as

If and only if

iffif, and only ifmaterial equivalence
Such a series converges if and only if the absolute value of the common ratio is less than one ( < 1).