How to Write a Rule for the Common Ratio of Geometric Sequences

Write the first term of your rule, or formula, followed by an equal sign (=). The first term of your formula is r. This r represents the common ratio of your geometric sequence. For example, "r =."

Write the variable a. This variable will help you represent a term in your geometric sequence. For example, "r = a."

Write a subscript n + 1 after the a. This n is the number of terms preceding your a term; the 1, added to n, represents the a term itself. If your sequence is 3, 9, 27, the n value of 27 is 2 because there are two terms, 3 and 9, before the 27, and 27 itself is term 3 (2 + 1 = 3). For example, you write, "r = a(n+1)." Note the parentheses mean the n + 1 expression is a subscript, that is, printed in a smaller font in front of and below the a term.

Write a division symbol (/) after the a(n + 1) term. For example, "r = a(n+1)/."

Write another variable a after the division symbol. This a will allow you to represent the first term to the left of the a(n + 1) term. For example, "r = a(n+1)/a."

Write a single subscript n after the a. Like the first subscript n you wrote, this subscript n represents the number of terms preceding this a term. In the geometric sequence 3, 9, 27, the n-value of 9 is 1 because there is only one term (3) in front of the 9. For example, you write, "r = a(n+1)/a(n)." Here too, the parentheses mean the n term is a subscript. The rule for the common ratio of a geometric sequence is r = a(n+1)/a(n).

Write an example calculation using your rule. For example, using the sequence 3, 9, 27, if your n value is 2, then a(n + 1) equals 27 because 27 is the third term (2 + 1 = 3), and a(n) = 9 because 9 is the second term (n = 2). You write, "r = 27/9." The common ratio (r) of your sequence is 27/9, or 3.