Geometric progression

A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2.

Examples of a geometric sequence are powers r^k of a fixed non-zero number r, such as 2^k and 3^k. The general form of a geometric sequence is

${\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots }$

where r is the common ratio and a is the initial value.

The sum of a geometric progression's terms is called a geometric series. Because each two successive numbers in the progression have the same proportion, the terms in a geometric series are also said to be in continued proportion, especially in the context of ancient Greek mathematics, where geometric progressions were described in this way rather than through powers of the common ratio.