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geometric sequence
Also known as a geometric progression, a finite sequence of at least three numbers, or an infinite sequence, whose terms differ by a constant multiple, known as the common ratio. For example, starting with 3 and using a common ratio of 2 leads to the finite geometric sequence: 3, 6, 12, 24, 48, and also to the infinite sequence 3, 6, 12, 24, 48, ..., (3 × 2n) ... In general, the terms of a geometric sequence have the form an = arn (n = 0, 1, 2, ...) for fixed numbers a and r. If the terms of a geometric sequence are added together the result is a geometric series. If it is a finite series, then we add its terms to get the series sum, Sn = a + ar + ar2 + ... + arn = (a - arn+1)/(1 - r). In the case of an infinite series, if |r| < 1, the sum is a/(1 - r). If |r| > or = 1, however, the series diverges and thus has no sum. Compare with arithmetic sequence.
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