Difference between revisions of "Geometric sequence"
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==Definition== | ==Definition== | ||
| − | A geometric sequence is a sequence of numbers | + | A geometric sequence is a sequence of numbers in which each term is a fixed multiple of the previous term. For example: 1, 2, 4, 8, 16, 32, ... is a geometric sequence because each term is twice the previous term. In this case, 2 is called the common ratio of the sequence. More formally, a geometric sequence may be defined [[recursion|recursively]] by: |
| − | <math>a_n = r | + | <math>a_n = r\cdot a_{n-1}, n \geq 1</math> |
| − | with a fixed <math> | + | with a fixed <math>a_0</math> and common ratio <math>r</math>. Using this definition, the <math>n</math>th term has the closed-form: |
| − | <math>\displaystyle a_n = | + | <math>\displaystyle a_n = a_0\cdot r^n</math> |
==Summing a Geometric Sequence== | ==Summing a Geometric Sequence== | ||
Revision as of 08:52, 23 June 2006
Definition
A geometric sequence is a sequence of numbers in which each term is a fixed multiple of the previous term. For example: 1, 2, 4, 8, 16, 32, ... is a geometric sequence because each term is twice the previous term. In this case, 2 is called the common ratio of the sequence. More formally, a geometric sequence may be defined recursively by:
with a fixed 
and common ratio 
. Using this definition, the 
th term has the closed-form:
Summing a Geometric Sequence
The sum of the first terms of a geometric sequence is given by
where 
is the first term in the sequence, and is the common ratio.
Infinate Geometric Sequences
An infinate geometric sequence is a geometric sequence with an infinate number of terms. These sequences can have sums, sometimes called limits, if 
.
For instance, the series 
, sums to 2. The general fromula for the sum of such a sequence is:
Again, is the first term in the sequence, and is the common ratio.
