# geometric sequence

A geometric sequence, also known as a **geometric progression**, is a finite sequence of at least three numbers, or an infinite sequence, whose terms differ by
a constant multiple, known as the **common ratio** (or common
quotient), *r*. A geometric sequence is uniquely determined by its
initial term and the ratio *r*.

For example, starting with 3 and using a common ratio of 2 leads to the
finite geometric sequence: 3, 6, 12, 24, 48, and also to the infinite sequence
3, 6, 12, 24, 48, ..., (3 × 2*n*) ...

In general, the terms of a geometric sequence have the form *a _{n}* =

*ar*(

^{n}*n*= 0, 1, 2, ...) for fixed numbers

*a*and

*r*.

In a geometric sequence, every term is the geometric
mean of its neighboring terms: *a _{n}* = √(

*a*). The geometric mean of any two different positive numbers is always less than their arithmetic mean.

_{n-1}× a_{n+1}

The following cases of geometric sequences can be distinguished:

1. *r* > 1.

The sequence increases, e.g., 1, 2, 4, 8, 16, ...

2. *r* = 1.

All terms of the sequence are equal, e.g., 2, 2, 2, ...

3. 0 < r < 1.

The sequence is decreasing, e.g., *a*_{1} = 2, *r* 1/2: 2, 1, 1/2, 1/4, 1/8, 1/16, ...

4. 0 > r > –1.

The terms of the sequence are alternately positive and negative, e.g., *a* = 1, *r* = –1/10: 1, –1/10, 1/100, –1/1000, ...

5. –1 > q.

The terms of the sequence have alternately positive and negative signs
and their absolute value increases,
e.g., *a* = 1, *r* = –2: 1, –2, +4, –8, +16, ...

Sequences with *r* > 1 and with –1 > *r* are **divergent
sequences**. Sequences with *r* = –1 are **oscillatory**.
Sequences with –1 < r ≤ +1 are **convergent**.

If the terms of a geometric sequence are added together the result is a **geometric series**. If it is a finite series, then the terms
are added to get the series sum, *S _{n}* =

*a*+

*ar*+

*ar*

^{2}+ ... +

*ar*= (

^{n}*a*–

*ar*

^{ n+1})/(1 –

*r*). In the case of an infinite series, if |

*r*| < 1, the sum is

*a*/(1 –

*r*). If |

*r*| > or = 1, however, the series diverges and thus has no sum.