Much like the arithmetic sequence, a geometric sequence has an initial term. However, a geometric sequence has a common ratio that yields the following number in the geometric sequence when multiplied times the previous number. For example, a geometric sequence with the common ratio of “4” would start like this: 1, 4, 16, 64, 256, …. Geometric sequences grow by multiplication. Geometric expansion is the most common mathematical model for epidemics because of its rapid growth. As with the arithmetic sequence, a geometric sequence may start with a negative initial value.

You find the common ratio, if one exists, by dividing this term by the preceding number and then verifying that this behavior is consecutive. In the example in the previous paragraph, we would divide 16 by 4, yielding 4. We would divide 64 by 16, producing 4. As we continue, we verify that the common ratio in this sequence is “4.”