How to Find the Geometric Sequence

By Michael Judge
The Greek mathematician Euclid wrote of geometric sequences in in 300 B.C.

In a geometric sequence, each number in a series of numbers is produced by multiplying the previous value by a fixed factor. If the first number in the series is "a" and the factor is "f," the series would be a, af, af^2, af^3 and so on. The ratio between any two adjacent numbers will give the factor. For example, in the series 2, 4, 8, 16 ... the factor is 16/8 or 8/4 = 2. A given geometric sequence is defined by its first term and the ratio factor, and these can be calculated if you are given enough information about that sequence.



Write down the information you are given about the sequence. You might be given the first term in the sequence ("a") and one or more consecutive numbers in the sequence. For instance, the first term could be 1 and the next term 2. Or you could be given any number in the progression, its position in the sequence and the ratio factor ("f"). An example would be that the second number in the sequence is 6 and the factor 2.

Divide the first term, a, into the second number in the sequence, when this is the information you are given. This will give you the ratio factor, f, for the sequence. In the example progression beginning with 1, 2, the factor would equal 2/1 = 2. The sequence is then defined as a succession of terms where each term equals (a)[f^(n - 1)] and n is the position of the term. So the fourth term in the example would be (1)[2^(4 - 1)] or 8. The sequence itself would be 1, 2, 4, 8, 16 ...

Calculate the first term in the sequence using the formula a = t/[f^(n - 1)], in cases where you are given a single number, t, and its position in the sequence, n, as well as the factor. So if the second term in the sequence (at n = 2) is 6 and f = 2, a = 6/[2^(2 - 1)] = 3. You now have the first term, 3, and the factor, 2, that define the sequence, so you can write the sequence as 3, 6, 12, 24 ...

Tip

Geometric sequences can be infinite or can have a defined number of terms. It is possible for the ratio factor to be less than one or negative, or both.

About the Author

Michael Judge has been writing for over a decade and has been published in "The Globe and Mail" (Canada's national newspaper) and the U.K. magazine "New Scientist." He holds a Master of Science from the University of Waterloo. Michael has worked for an aerospace firm where he was in charge of rocket propellant formulation and is now a college instructor.