Geometric Progression

Geometric Progression, in mathematics, sequence of numbers in which the ratio of any term, after the first, to the preceding term is a fixed number, called the common ratio. For example, the sequence of numbers 2, 4, 8, 16, 32, 64, 128 is a geometric progression in which the common ratio is 2, and 1, €, Š, }, ‰, |, ...€i, ... is a geometric progression in which the common ratio is €. The first is a finite geometric progression with seven terms; the second is an infinite geometric progression. In general, a geometric progression may be described by denoting the first term in the progression by a, the common ratio by r, and, in a finite progression, the number of terms by n. A finite geometric progression may then be written formally as

and an infinite geometric progression as

In general, if the nth term of a geometric progression is denoted by a_n, it follows from the definition that

If the symbol S_n denotes the sum of the first n terms of a geometric progression, it can be proved that

The terms in a geometric progression between a_i, and a_j, i<j, are called geometric means. The geometric mean between two positive numbers x and y is the same as the mean proportional Ë between the two numbers. In particular, a_n is the geometric mean or mean proportional between a_{n - 1} and a_{n + 1}.

The formal sum of the terms of an infinite geometric progression, written as

is called a geometric series (see Sequence and Series). In analysis it can be proved that a geometric series converges if the absolute value of the common ratio is less than 1; otherwise, the series diverges. If the series does converge, the limit, S, can be shown to be

The symbol is read “the limit of S_n as n increases without bound.”

Geometric series and geometric progressions have many applications in the physical, biological, and social sciences, as well as in investments and banking. Many problems in compound interest and annuities are easily solved using these concepts.

See also Arithmetic Progression.