# 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
145 Related Articles

### 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.

### Harmonic progression (mathematics)

Harmonic progressionharmonic sequence

### 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.

### Extended real number line

extended real numbersextended real line+∞

### Constant function

constantconstant mapconstant mapping

### Point at infinity

points at infinityat infinityinfinity

### 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).