Difference between revisions of "Geometric sequence"
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An infinate geometric sequence is a geometric sequence with an infinate number of terms. These sequences can have sums, sometimes called limits, if <math>|r|<1</math>. | An infinate geometric sequence is a geometric sequence with an infinate number of terms. These sequences can have sums, sometimes called limits, if <math>|r|<1</math>. | ||
| − | For instance, the series <math>1 + \frac12 + \frac14 + \frac18 + | + | For instance, the series <math>1 + \frac12 + \frac14 + \frac18 + \cdots</math>, sums to 2. The general fromula for the sum of such a sequence is: |
<math>S = \frac{a_1}{1-r}</math> | <math>S = \frac{a_1}{1-r}</math> | ||
Revision as of 04:20, 23 June 2006
Definition
A geometric sequence is a sequence of numbers where the nth term of the sequence is a 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 as:
with a fixed 
and common ratio 
. Using this definition, the 
th term may be found explicityly with:
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.
