How to Find the Geometric Sequence

The Greek mathematician Euclid wrote of geometric sequences in in 300 B.C.
••• Images

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 ...


    • 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.

Related Articles

How to Factor Monomials
Definition of Successor and Predecessor in Math
What is the Difference Between a Sequence and a Series?
How to Write Polynomial Functions When Given Zeros
How to Find Fraction Sequences
How to Solve an Arithmetic Sequence Problem With Variable...
How to Find All The Factors of a Number Quickly and...
What is a Geometric Sequence?
How to Find the Cross Sectional Area of a Pipe
How to Find the Line of Symmetry in a Quadratic Equation
How to Find Factors of a Constant Term
Characteristics of Aquatic Plants
How to Find the Common Ratio of a Fraction
How to Calculate the Sum of a Geometric Series
How to Calculate With the Taylor Series
How to Factor Polynomials With 4 Terms
How to Find Equations of Tangent Lines
How to Calculate Moles from Molecular Weight
How to Calculate Arctan
How to Convert a Fraction to a Ratio