# Geometric sequence

## Contents

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

• 3,6,12,24,48,96 (finite, length 6: 6 elements, quotient 2)
• 1, − 2,4, − 8 (finite, length 4: 4 elements, quotient −2)
• $8, 4, 2, 1, {1\over2}, {1\over4}, {1\over8}, \dots {1\over2^{n-4}}, \dots$ (infinite, quotient $1\over2$)
• $2, 2, 2, 2, \dots$ (infinite, quotient 1)
• $-2, 2, -2, 2, \dots , (-1)^n\cdot 2 , \dots$ (infinite, quotient −1)
• ${1\over2}, 1, 2, 4, \dots , 2^{n-2}, \dots$ (infinite, quotient 2)
• $1, 0, 0, 0, \dots \$ (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 An of the deposit after n time-periods is given by

$A_n = A \left( 1 + {p\over100} \right)^n$

i.e., the values A=A0, A1, A2, A3, ... form a geometric sequence with quotient q = 1+(p/100).

##  Mathematical notation

A finite sequence

$a_1,a_2,\dots,a_n = \{ a_i \mid i=1,\dots,n \} = \{ a_i \}_{i=1,\dots,n}$

or an infinite sequence

$a_0,a_1,a_2,\dots = \{ a_i \mid i\in\mathbb N \} = \{ a_i \}_{i\in\mathbb N}$

is called geometric sequence if

${ a_{i+1} \over a_i } = q$

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

ai = a1qi − 1

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

• a1 = 0 , q arbitrary: 0, 0•q = 0, 0, 0, ...
• q = 0 : a1, 0•a1 = 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

$a_1 + a_2 +\cdots+ a_n = \sum_{i=1}^n a_i$
$= a_1 ( 1+q+q^2+ \cdots +q^{n-1} ) = \begin{cases} a_1 { 1-q^n \over 1-q } & q \ne 1 \\ a_1 \cdot n & q = 1 \end{cases}$

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

$\sum_{i=0}^\infty a_0 q^i = a_0 { 1 \over 1-q } \qquad (\textrm {for}\ |q|<1)$