Geometric progression
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 rk of a fixed non-zero number r, such as 2k and 3k.
The general form of a geometric sequence is where r is the common ratio and a is the initial value.
The nth term of a geometric sequence with initial value a = a1 and common ratio r is given by and in general Geometric sequences satisfy the linear recurrence relation This is a first order, homogeneous linear recurrence with constant coefficients.
Geometric sequences also satisfy the nonlinear recurrence relation
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 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.
Malthus as the mathematical foundation of his An Essay on the Principle of Population.
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.
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.
The partial product of a geometric progression up to the term with power
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
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.
Sums of logarithms correspond to products of exponentiated values.
Written out in full, Carrying out the multiplications and gathering like terms, The exponent of r is the sum of an arithmetic sequence.
Substituting the formula for that sum, which concludes the proof.
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 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.
