Difference between revisions of "Geometric sequence"
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The sum of the first <math>n</math> terms of a geometric sequence is given by | The sum of the first <math>n</math> terms of a geometric sequence is given by | ||
| − | <math>S_n = a_0 + a_1 + \ldots + a_{n - 1} = a_0\cdot\frac{r^n- | + | <math>S_n = a_0 + a_1 + \ldots + a_{n - 1} = a_0\cdot\frac{r^n-1}{r-1}</math> |
where <math>a_0</math> is the first term in the sequence, and <math>r</math> is the common ratio. | where <math>a_0</math> is the first term in the sequence, and <math>r</math> is the common ratio. | ||
Revision as of 08:54, 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.
