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 −2)

• $8,4,2,1,{1 \over 2},{1 \over 4},{1 \over 8},\dots {1 \over 2^{n-4}},\dots$ (infinite, quotient $1 \over 2$)

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

• $-2,2,-2,2,\dots ,(-1)^{n}\cdot 2,\dots$ (infinite, quotient −1)

• ${1 \over 2},1,2,4,\dots ,2^{n-2},\dots$ (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 A_n of the deposit after n time-periods is given by

$A_{n}=A\left(1+{p \over 100}\right)^{n}$

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:

• 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

$a_{1}+a_{2}+\cdots +a_{n}=\sum _{i=1}^{n}a_{i}$

$=a_{1}(1+q+q^{2}+\cdots +q^{n-1})={\begin{cases}a_{1}{1-q^{n} \over 1-q}&q\neq 1\\a_{1}\cdot n&q=1\end{cases}}$

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

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