Geometric sequence
From Wikinfo
In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant.
For example, the sequence
- 1, 2, 4, 8, 16, 32, ....
in which each number is multiplied by 2, which is the common ratio, to get the next number, is a geometric sequence. The sequence
- 729, 486, 324, 216, 144, 96, 64
is a geometric sequence whose common ratio is 2/3, i.e., each term is multiplied by 2/3 to get the next term. The sequence
- 3, −3, 3, −3, 3, −3, 3, −3, 3, −3, ...
is a geometric sequence in which the common ratio is −1.
A geometric progression has exponential growth or exponential decay.
If the initial term of a geometric progression is a and the common quotient of successive members is r, then the n-th term of the sequence is given by a rn, n = 0, 1, 2, ...
The sum of the numbers in a geometric progression is called a geometric series. A convenient formula for geometric series is available:
- <math>\sum_{k=0}^{n-1} a\,r^k=a\frac{r^{n}-1}{r-1}</math>
Compare this with a arithmetic progression showing linear growth (or decline) such as 4, 15, 26, 37, 48, .... Note that the two kinds of progression are related: taking the logarithm of each term in a geometric progression yields an arithmetic one.
References
- Adapted from the Wikipedia article, "Geometric_sequence" http://en.wikipedia.org/wiki/Geometric_sequence, used under the GNU Free Documentation License
