What is a Geometric Sequence?

In mathematics, sequences are ordered lists of numbers that follow a pattern. You have probably encountered arithmetic sequences before — where you add or subtract the same number each time. A geometric sequence is different: instead of adding, you multiply by the same number each step.

Geometric Sequence

A geometric sequence is an ordered list of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Here are some examples to get a feel for them:

2, 6, 18, 54, 162, … (multiplying by 3 each time)
100, 50, 25, 12.5, … (multiplying by 0.5 each time)
1, −2, 4, −8, 16, … (multiplying by −2 each time)
81, 27, 9, 3, 1, … (multiplying by $\frac{1}{3}$ each time)

Notice that geometric sequences can grow (when the ratio is greater than 1), shrink (when the ratio is between 0 and 1), or even alternate in sign (when the ratio is negative).

Analogy

Think of a geometric sequence like a chain reaction. If one person tells 3 friends, and each of those friends tells 3 more friends, the number of people who know keeps tripling: 1, 3, 9, 27, 81… That's a geometric sequence with a common ratio of 3!

The Common Ratio

Common Ratio

The common ratio (denoted $r$ ) is the fixed multiplier between consecutive terms in a geometric sequence. It is found by dividing any term by the term immediately before it:
$r = \frac{T _{n + 1}}{T _{n}}$

To check whether a sequence is geometric, divide consecutive pairs of terms. If you always get the same value, the sequence is geometric and that value is $r$ .

Example

Example: Finding the common ratio

Is the sequence 5, 15, 45, 135, 405 geometric? If so, find $r$ .

Step 1: Divide consecutive terms:
$\frac{15}{5} = 3, \frac{45}{15} = 3, \frac{135}{45} = 3, \frac{405}{135} = 3$

Step 2: Since all ratios are equal, yes — this is a geometric sequence with $r = 3$ .

Example

Example: Identifying a non-geometric sequence

Is the sequence 2, 4, 8, 11 geometric?

$\frac{4}{2} = 2, \frac{8}{4} = 2, \frac{11}{8} = 1.375$

The ratios are not equal, so this is not a geometric sequence.

Warning

Do not confuse geometric sequences with arithmetic sequences. In arithmetic sequences you add a common difference; in geometric sequences you multiply by a common ratio. The sequence 2, 4, 6, 8 is arithmetic (add 2), while 2, 4, 8, 16 is geometric (multiply by 2).

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