## geometric sequenceA geometric sequence, also known as a geometric progression, is a finite sequence of at least three numbers, or an infinite sequence, whose terms differ by
a constant multiple, known as the common ratio (or common
quotient), r. A geometric sequence is uniquely determined by its
initial term and the ratio r. 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 × 2 n) ... In general, the terms of a geometric sequence have the form a = _{n}ar (^{n}n = 0, 1, 2, ...) for fixed numbers a and r. In a geometric sequence, every term is the geometric mean of its neighboring terms: a = √(_{n}a). The geometric mean of any two different positive
numbers is always less than their arithmetic
mean. _{n-1} × a_{n+1}The following cases of geometric sequences can be distinguished: *r*> 1.
The sequence increases, e.g., 1, 2, 4, 8, 16, ...-
*r*= 1.
All terms of the sequence are equal, e.g., 2, 2, 2, ... - 0 < r < 1.
The sequence is decreasing, e.g.,*a*_{1}= 2,*r*1/2: 2, 1, 1/2, 1/4, 1/8, 1/16, ... - 0 > r > -1.
The terms of the sequence are alternately positive and negative, e.g.,*a*= 1,*r*= -1/10: 1, -1/10, 1/100, -1/1000, ... - -1 > q.
The terms of the sequence have alternately positive and negative signs and their absolute value increases, e.g.,*a*= 1,*r*= -2: 1, -2, +4, -8, +16, ...
Sequences with r > 1 and with -1 > r are divergent
sequences. Sequences with r = -1 are oscillatory.
Sequences with -1 < r ≤ +1 are convergent. If the terms of a geometric sequence are added together the result is a geometric series. If it is a finite series, then the terms
are added to get the series sum, S = _{n}a + ar + ar ^{2} + ... + ar = (^{n}a - ar^{ n+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. ## Related entries• arithmetic sequence• harmonic sequence ## Related category• SERIES AND SEQUENCES | |||||

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