Geometric progression

A sequence of numbers each one of which is equal to the preceding one multiplied by a number (the denominator of the progression). A geometric progression is called increasing if , and decreasing if ; if , one has a sign-alternating progression. Any term of a geometric progression can be expressed by its first term and the denominator by the formula

The sum of the first terms of a geometric progression (with ) is given by the formula

If , the sum tends to the limit as increases without limit. This number is known as the sum of the infinitely-decreasing geometric progression.

The expression

is the simplest example of a convergent series — a geometric series; the number is the sum of the geometric series.

The term "geometric progression" is connected with the following property of any term of a geometric progression with positive terms: , i.e. any term is the geometric mean of the term which precedes it and the term which follows it.