# Geometric sequence

A **geometric sequence** (or **geometric progression**) is a (finite or infinite) sequence
of (real or complex) numbers
such that the quotient (or ratio) of consecutive elements is the same for every pair.

In finance, compound interest generates a geometric sequence.

## Examples

Examples for geometric sequences are

- (finite, length 6: 6 elements, quotient 2)

- (finite, length 4: 4 elements, quotient −2)

- (infinite, quotient )

- (infinite, quotient 1)

- (infinite, quotient −1)

- (infinite, quotient 2)

- (infinite, quotient 0) (See General form below)

## Application in finance

The computation of compound interest leads to a geometric series:

When an initial amount *A* is deposited at an interest rate of *p* percent per time period
then the value *A*_{n} of the deposit after *n* time-periods is given by

i.e., the values
*A*=*A*_{0}, *A*_{1}, *A*_{2}, *A*_{3}, ...
form a geometric sequence with quotient *q* = 1+(*p*/100).

## Mathematical notation

A finite sequence

or an infinite sequence

is called geometric sequence if

for all indices *i* where *q* is a number independent of *i*. (The indices need not start at 0 or 1.)

### General form

Thus, the elements of a geometric sequence can be written as

**Remark:** This form includes two cases not covered by the initial definition depending on the quotient:

*a*_{1}= 0 ,*q*arbitrary: 0, 0•*q*= 0, 0, 0, ...*q = 0*:*a*_{1}, 0•*a*_{1}= 0, 0, 0, ...

(The initial definition does not cover these two cases because there is no division by 0.)

### Sum

The sum (of the elements) of a finite geometric sequence is

The sum of an infinite geometric sequence is a geometric series: