Geometric sequence: Difference between revisions
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== Application in finance == | == 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''<sub>n</sub> 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''=''A''<sub>0</sub>, ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, ... | |||
form a geometric sequence with quotient ''q'' = 1+(''p''/100). | |||
== Mathematical notation == | == Mathematical notation == |
Revision as of 18:45, 17 January 2010
A geometric sequence is a (finite or infinite) sequence of (real or complex) numbers such that the quotient 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 )
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. (The indices need not start at 0 or 1.)
General form
Thus, the elements of a geometric sequence can be written as
Sum
The sum (of the elements) of a finite geometric sequence is
The sum of an infinite geometric sequence is a geometric series: