Common Ratio Formula
Before learning the common ratio formula, let us recall what is the common ratio. The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio", because if you divide (that is, if you find the ratio of) successive terms, you'll always get this value. The common ratio formula helps in calculating the common ratio for a given geometric progression.
What Is the Common Ratio Formula?
The general form of representing a geometric progression is a_{1}, (a_{1}r), (a_{1}r^{2}), (a_{1}r^{3}), (a_{1}r^{4}) ,... where a_{1 }is the first term of GP, a_{1}r is the second term of GP, and r is the common ratio. Thus, the common ratio formula of a geometric progression is given as:
Common ratio, \(r = \frac{a_n}{a_{n1}}\)
where,
 a\(_n\) is the n^{th }term of the geometric progression.
 a\(_n1\) is the (n  1)^{th }term of the geometric progression.
Let us see the applications of the common ratio formula in the following section.

Example 1: Find the common ratio for the geometric sequence 1, 2, 4, 8, 16,... using the common ratio formula.
Solution:
To find: Common ratio
Divide each term by the previous term to determine whether a common ratio exists.
\(\frac{2}{1} = \frac{4}{2} = \frac{8}{4} = \frac{16}{8} = 2 \)
The sequence is geometric because there is a common multiple, 2, which is called the common ratio.
Answer: Common ratio, r = 2

Example 2: What is the common ratio for a geometric sequence whose formula is: a\(_n\) = 4(3)^{n1}?
Solution:
To find: Common ratio
Given: Formula of geometric sequence = 4(3)^{n1}
A listing of the terms will show what is happening in the sequence (start with n = 1).
4, 12, 36,108,...
Using the common ratio formula,
r = 12/4 =3
Answer: Common ratio, r = 3