Geometric sequence

Geometric sequence

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.

Examples of a geometric sequence are powers r^k of a fixed number r, such as 2^k and 3^k. The general form of a geometric sequence is

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

Diagram illustrating three basic geometric sequences of the pattern 1(rⁿ⁻¹) up to 6 iterations deep.

The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.

Video: Extending geometric sequences

Elementary properties

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

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 sequence 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

The behaviour of a geometric sequence depends on the value of ....

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