Sign In
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.
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.
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, ...