Geometric sequence: Difference between revisions

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imported>Peter Schmitt
(→‎Sum: (blushing) taking care of q=1)
imported>Boris Tsirelson
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=== Sum ===
=== Sum ===
The sum (of the elements) of a finite geometric sequence is
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> a_1 + a_2 +\cdots+ a_n = \sum_{i=1}^n a_i </math>
      = a_1 ( 1+q+q^2+ \cdots +q^{n-1} )
: <math> = a_1 ( 1+q+q^2+ \cdots +q^{n-1} )
       = \begin{cases}  a_1 { 1-q^n  \over 1-q } & q \ne 1 \\
       = \begin{cases}  a_1 { 1-q^n  \over 1-q } & q \ne 1 \\
                         a_1 \cdot n              & q = 1  
                         a_1 \cdot n              & q = 1  

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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

  • (finite, length 6: 6 elements, quotient 2)
  • (finite, length 4: 4 elements, quotient −2)
  • (infinite, quotient )
  • (infinite, quotient 1)
  • (infinite, quotient −1)
  • (infinite, quotient 2)
  • (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

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

Mathematical notation

A finite sequence

or an infinite sequence

is called geometric sequence if

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

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

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