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

Thus, the general form of a geometric sequence is

and that of a geometric series is

where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.

(Our solved example in mathguru.com uses this concept)

The n-th term of a geometric sequence with initial value a and common ratio r is given by

Such a geometric sequence also follows the recursive relation

 for every integer 

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio

The common ratio of a geometric series may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance

1, −3, 9, −27, 81, −243, ...

is a geometric sequence with common ratio −3.

http://en.wikipedia.org/wiki/Geometric_progression

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