Geometric sequence
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 called the common ratio of the sequence.
Thus without loss of generality a geometric sequence can be written as <math>a(r^0,r^1,r^2,r^3,...)\,<math> where r is the common ratio and a is a scale factor. Thus the common ratio gives a family of geometric sequences whose starting value is determined by the scale factor.
For example, a sequence with a common ratio of 2 and a scale factor of 1 is
- 1, 2, 4, 8, 16, 32, ....
and a sequence with a common ratio of 2/3 and a scale factor of 729 is
- 729 (1, 2/3, 4/9, 8/27, 16/81, 32/243, 64/729, ....) = 729, 486, 324, 216, 144, 96, 64, ....
and finally a sequence with a common ratio of -1 and a scale factor of 3 is
- 3 (1, −1, 1, −1, 1, −1, 1, −1, 1, −1, ....) = 3, −3, 3, −3, 3, −3, 3, −3, 3, −3, ....
The sum of the numbers in a geometric progression is called a geometric series. Thus the geometric series for the n terms of a geometric progression is
<math>a(r^0+r^1+r^2+r^3+...+r^{n-1})\,<math>
Multiplying <math>(1+r+r^2+r^3+...+r^{n-1})\,<math> by <math>(1-r)\,<math> equals <math>(1-r^n)\,<math> since all the other terms cancel in pairs.
Rearranging gives the convenient formula for a geometric series:
- <math>a\sum_{k=0}^{n-1} r^k=a\frac{r^{n}-1}{r-1}<math>
An interesting relationship for a geometric series is given by:
<math>a(r^0+r^1+r^2+r^3+r^4+r^5+r^6+r^7+r^8+...) = a(r^0+r^1+r^2)(r^0+r^3+r^6+...)\,<math>
For example, (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 +...) can be written as (1 + 2 + 4)(1 + 8 + 64 +...)
A geometric progression has exponential growth or exponential decay.
Compare this with an arithmetic progression showing logarithm of each term in a geometric progression yields an arithmetic one.
