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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

  •  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)

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:

  • 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)
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