Geometric sequence

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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

  • <math> 3, 6, 12, 24, 48, 96 </math> (finite, length 6: 6 elements, quotient 2)
  • <math> 1, -2, 4, -8 </math> (finite, length 4: 4 elements, quotient −2)
  • <math> 8, 4, 2, 1, {1\over2}, {1\over4}, {1\over8},
                    \dots {1\over2^{n-4}}, \dots  </math> (infinite, quotient <math>1\over2</math>)
  • <math> 2, 2, 2, 2, \dots </math> (infinite, quotient 1)
  • <math> -2, 2, -2, 2, \dots , (-1)^n\cdot 2 , \dots </math> (infinite, quotient −1)
  • <math> {1\over2}, 1, 2, 4, \dots , 2^{n-2}, \dots </math> (infinite, quotient 2)
  • <math> 1, 0, 0, 0, \dots \ </math> (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

<math> A_n = A \left( 1 + {p\over100} \right)^n </math>

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

Mathematical notation

A finite sequence

<math> a_1,a_2,\dots,a_n = \{ a_i \mid i=1,\dots,n \}
        = \{ a_i \}_{i=1,\dots,n} </math>

or an infinite sequence

<math> a_0,a_1,a_2,\dots = \{ a_i \mid i\in\mathbb N \}
        = \{ a_i \}_{i\in\mathbb N} </math>

is called geometric sequence if

<math> { a_{i+1} \over a_i } = q </math>

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

<math> a_i = a_1 q^{i-1} </math>

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

<math> a_1 + a_2 +\cdots+ a_n = \sum_{i=1}^n a_i </math>
<math> = 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}

</math>

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

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