Definition and reason for a geometric sequence

What is a geometric sequence?

A sequence (u_n) is said to be geometric if, to move from one term to the next, we always multiply by the same number. This number is called the common ratio, denoted by q.

In other words, for any integer n:

u_(n+1) = u_n * q

Practical example

Consider the sequence 3, 6, 12, 24, 48, ... We can see that each term is twice the previous one: 6/3 = 2, 12/6 = 2, 24/12 = 2. The common ratio is therefore q = 2, and the first term is u_0 = 3.

n	0	1	2	3	4
u_n	3	6	12	24	48

How do you recognise a geometric sequence?

Calculate the ratio u_(n+1)/u_n for several values of n. If it is constant, the sequence is geometric and this ratio is the common ratio q.

Note: if the ratio changes from one term to the next, the sequence is not geometric (it may be arithmetic, or of another type).

Common pitfall

Do not confuse the geometric common ratio (multiplication) with the arithmetic common ratio (addition). In an arithmetic sequence, a constant r is ADDED; in a geometric sequence, the terms are MULTIPLIED by a constant q. Always check by division, not subtraction, before drawing any conclusions.