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


 * $$ 3, 6, 12, 24, 48, 96           $$ (finite, length 6: 6 elements, quotient 2)


 * $$ 1, -2, 4, -8                   $$ (finite, length 4: 4 elements, quotient &minus;2)

\dots {1\over2^{n-4}}, \dots $$ (infinite, quotient $$1\over2$$)
 * $$ 8, 4, 2, 1, {1\over2}, {1\over4}, {1\over8},


 * $$ 2, 2, 2, 2, \dots $$ (infinite, quotient 1)


 * $$ -2, 2, -2, 2, \dots, (-1)^n\cdot 2 , \dots $$ (infinite, quotient &minus;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
 * $$ a_i = a_1 q^{i-1} $$

Remark: This form includes two cases not covered by the initial definition depending on the quotient: (The initial definition does not cover these two cases because there is no division by 0.)
 * a1 = 0, q arbitrary: 0, 0•q = 0, 0, 0, ...
 *  q = 0 : a1, 0•a1 = 0, 0, 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) $$