Geometric Sequences
Concept

Geometric Sequence

A geometric sequence is a sequence in which the ratio r between consecutive terms is a nonzero constant. This ratio is called the common ratio. The following is an example geometric sequence with first term 3 and common ratio 2.
Geometric Sequence: 3, 6, 12, 24, 48, ... with a common ratio of 2
The behavior of geometric sequences generally depends on the values of the first term a_1 and the common ratio r. The following table shows the effects of these parameters.
a_1> 0 a_1< 0
r>1 Increasing
3 * 2 → 6 * 2 → 12 * 2 → 24 * 2 → 48 ...
Decreasing
- 3 * 2 → - 6 * 2 → - 12 * 2 → - 24 * 2 → - 48 ...
r=1 Constant

3 * 1 → 3 * 1 → 3 * 1 → 3 * 1 → 3 ...

Constant

- 3 * 1 → - 3 * 1 → - 3 * 1 → - 3 * 1 → - 3 ...

0 < r < 1 Decreasing
48 * 12 → 24 * 12 → 12 * 12 → 6 * 12 → 3 ...
Increasing
- 48 * 12 → - 24 * 12 → - 12 * 12 → - 6 * 12 → - 3 ...
r < 0 Alternating

3 * (- 2) → - 6 * (- 2) → 12 * (- 2) → - 24 * (- 2) → 48 ...

Alternating

- 3 * (- 2) → 6 * (- 2) → - 12 * (- 2) → 24 * (- 2) → - 48 ...

Like for any other sequence, the first term of a geometric sequence is denoted by a_1, the second by a_2, and so on. Since geometric sequences have a common ratio r, once one term is known, the next term can always be found by multiplying the known term by r.

General geometric sequence a_1, a_2, a_3, a_4, a_5, ... with a common ratio of r.

In fact, the sequence can be found using only a_1 and r, since all the subsequent terms can be found by multiplying a_1 by r a specific number of times. Because of this, geometric sequences have the following general form.


a_1, a_1r, a_1r^2, a_1r^3, a_1r^4, ...

Exercises