geometric progression

In mathematics, a sequence of terms (progression) in which each term is a constant multiple (called the common ratio) of the one preceding it. For example, 3, 12, 48, 192, 768, ... is a geometric progression with a common ratio 4, since each term is equal to the previous term multiplied by 4. Compare arithmetic progression.

The sum of n terms of a geometric series

1 + r + r2 + r3 + ... + rn − 1

is given by the formula

Sn = (1 − rn)/(1 − r) for all r ≠ 1.

For mod r < 1 (see absolute value), the geometric series can be summed to infinity:

S∞ = 1/(1 − r).

In nature, many single-celled organisms reproduce by splitting in two so that one cell gives rise to 2, then 4, then 8 cells, and so on, forming a geometric sequence 1, 2, 4, 8, 16, 32, ..., in which the common ratio is 2.