A 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 × 2n) ...

 

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_n-1 × a_n+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 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.