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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 (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 × 2n) ...

In general, the terms of a geometric sequence have the form an = 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: an = √(an-1 × an+1). The geometric mean of any two different positive numbers is always less than their arithmetic mean.

The following cases of geometric sequences can be distinguished:
  1. r > 1.

    The sequence increases, e.g., 1, 2, 4, 8, 16, ...

  2. r = 1.

    All terms of the sequence are equal, e.g., 2, 2, 2, ...

  3. 0 < r < 1.

    The sequence is decreasing, e.g., a1 = 2, r 1/2: 2, 1, 1/2, 1/4, 1/8, 1/16, ...

  4. 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, ...

  5. -1 > q.

    The terms of the squence 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, Sn = 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