In mathematics, a geometric progression, also known as a geometric sequence, is a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

of non-zero

number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ...

s where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are

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''r''^''k'' of a fixed non-zero number ''r'', such as 2^''k'' and 3^''k''. The general form of a geometric sequence is :

a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots

where ''r'' ≠ 0 is the common ratio and ''a'' ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression terms is called a ''

geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each su ...

''.

Elementary properties

The ''n''-th term of a geometric sequence with initial value ''a'' = ''a''₁ and common ratio ''r'' is given by :

a_n = a\,r^,

and in general :

a_n = a_m\,r^.

Such a geometric sequence also follows the recursive relation :

a_n = r\,a_

for every integer

n\geq 2.

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers alternating between positive and negative. For instance :1, −3, 9, −27, 81, −243, ... is a geometric sequence with common ratio −3. The be