Difference between revisions of "Geometric sequence"
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| − | [[arithmetic sequence|Arithmetic Sequences]] | + | *[[arithmetic sequence|Arithmetic Sequences]] |
| + | *[[geometric series|Geometric Series]] | ||
| + | *[[sequence|Sequence]] | ||
| + | *[[series|Series]] | ||
Revision as of 09:04, 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.
Infinite Geometric Sequences
An infinite geometric sequence is a geometric sequence with an infinite number of terms. If the common ratio is small, the sum of the terms will approach a fixed limit. In this case, "small" means 
. We say that the sum of the terms of this sequence is a convergent sum.
For instance, the series 
, sums to 2. The general formula for the sum of such a sequence is:
Again, is the first term in the sequence, and is the common ratio.
One common instance of summing infinite geometric sequences is the decimal expansion of most rational numbers. For instance, 
has first term 
and common ratio 
, so the infinite sum has value 
, just as we would have expected.
