It's time for a few definitions.
Geometric Sequence
A sequence is an ordered list of numbers. A sequence can be either arithmetic or geometric. A sequence of values follows a pattern. In the case of a geometric sequence, each term is multiplied by a constant amount to arrive at the next term in the sequence.
The fixed amount that each term is multiplied by is called the common ratio and is denoted by r.
For example, 2, 4, 8, 16, ... is a geometric sequence with a common ratio of 2.
2 × 2 = 4
4 × 2 = 8
8 × 2 = 16
The common ratio can also be negative and/or a number less than 1. For example, 1, 0.75, .5625, 0.421875, ... is a geometric sequence with a common ratio of 0.75.
1 × 0.75 = 0.75
0.75 × 0.75 = 0.5625
0.5625 × 0.75 = 0.421875
Let's practice some more with geometric sequences. Look at the sequence and figure out its common ratio. Then use that to find the next term in the sequence. Click the header in each column to check your answers.
| Sequence | Common Ratio | Next Term |
| 3, 9, 27, 81, ... |
3 | 81 × 3 = 243 |
| Sequence | Common Ratio | Next Term |
| 2, 12, 72, 432, ... |
6 | 432 × 6 = 2592 |
| Sequence | Common Ratio | Next Term |
| \(\mathsf{ \frac{7}{5}, \frac{7}{15}, \frac{7}{45}, \frac{7}{135} }\), ... |
\(\mathsf{ \frac{1}{3} }\) | \(\mathsf{ \frac{7}{135} \times \frac{1}{3} = \frac{7}{405} }\) |
