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Definition of Geometric Sequences

The geometric sequence is another basic type of sequence. Understanding its definition is the first step toward mastering the laws of exponential growth.

Definition

Definition of a geometric sequence

If, starting from the second term, the ratio of each term to its predecessor in a sequence is always the same constant, then the sequence is called a geometric sequence. This constant is called the common ratio of the geometric sequence, usually denoted by the letter $q$ .

符号说明

符号	类型	读音/说明	在本文中的含义
$q$	参数	quotient	The common ratio of a geometric sequence, the ratio between adjacent terms

In mathematical language: for a sequence $\{a_n\}$ , if

$\frac{a_{n+1}}{a_n} = q \quad (n \in \mathbb{N}^*, q \neq 0 \text{ is a constant})$

then $\{a_n\}$ is a geometric sequence.

Note: every term of a geometric sequence must be nonzero; otherwise the ratio is meaningless.

Methods of Determination

There are several methods to determine whether a sequence is a geometric sequence:

Method 1: The Definition Method

Check whether the ratio between adjacent terms is constant.

Example 1: Determine whether the sequence $2, 6, 18, 54, \ldots$ is a geometric sequence.

Solution:

$\frac{a_2}{a_1} = \frac{6}{2} = 3$
$\frac{a_3}{a_2} = \frac{18}{6} = 3$
$\frac{a_4}{a_3} = \frac{54}{18} = 3$

The ratio between adjacent terms is always 3, so this is a geometric sequence with common ratio 3.

Method 2: The General Term Method

If the general term formula of a sequence can be written as $a_n = c \cdot q^n$ or $a_n = c \cdot q^{n-1}$ (where $c, q$ are nonzero constants), then the sequence is a geometric sequence.

Example 2: Is the sequence $a_n = 3 \cdot 2^{n-1}$ a geometric sequence?

Solution:

This is of the form $c \cdot q^{n-1}$ with $c = 3, q = 2$ , so it is a geometric sequence with common ratio $q = 2$ .

Verification: $a_1 = 3, a_2 = 6, a_3 = 12, \ldots$ , indeed a geometric sequence with common ratio 2.

The Meaning of the Common Ratio

The common ratio $q$ determines the trend of the geometric sequence:

$q > 1$ : the sequence is increasing (e.g., $2, 4, 8, 16, \ldots$ )
$0 < q < 1$ : the sequence is decreasing (e.g., $8, 4, 2, 1, \ldots$ )
$q < 0$ : the sequence alternates in sign (e.g., $1, -2, 4, -8, \ldots$ )
$q = 1$ : the sequence is constant (e.g., $5, 5, 5, 5, \ldots$ )

What is the essential difference between a geometric sequence and an arithmetic sequence?

An arithmetic sequence lives in the world of “addition”: each time a fixed amount (the common difference) is added, reflecting linear growth. Its general term formula $a_n = a_1 + (n-1)d$ is a linear function.

A geometric sequence lives in the world of “multiplication”: each time a fixed multiple (the common ratio) is applied, reflecting exponential growth. Its general term formula $a_n = a_1 \cdot q^{n-1}$ is an exponential function.

Figuratively speaking:

An arithmetic sequence is like climbing stairs, rising a fixed height each time
A geometric sequence is like a rolling snowball, growing by a fixed multiple each time

This is why geometric sequences grow faster (when $q > 1$ )!

Special Cases

A Sequence with Only One Term

A sequence with only one term can be regarded as a geometric sequence with any common ratio, because there are no “adjacent terms” to determine the common ratio.

A Sequence with Only Two Terms

Any two-term sequence (both nonzero) is a geometric sequence, with common ratio $q = \frac{a_2}{a_1}$ .

A Sequence Containing Zero

If a sequence contains a zero term, then it is not a geometric sequence, because the ratio is meaningless.

Practice Problems

Exercise 1

Determine whether the following sequences are geometric sequences. If so, state the common ratio:

$3, 9, 27, 81, \ldots$
$1, 1, 1, 1, \ldots$
$1, 2, 3, 4, \ldots$
$16, 8, 4, 2, \ldots$

Reference Answer(2 个标签)

geometric sequencecommon ratio

Idea: Check whether the ratio between adjacent terms is constant.

Detailed steps:

$\frac{a_2}{a_1} = \frac{9}{3} = 3$ , $\frac{a_3}{a_2} = \frac{27}{9} = 3$ , $\frac{a_4}{a_3} = \frac{81}{27} = 3$

It is a geometric sequence with common ratio $q = 3$
$\frac{a_2}{a_1} = \frac{1}{1} = 1$ , $\frac{a_3}{a_2} = \frac{1}{1} = 1$

It is a geometric sequence with common ratio $q = 1$ (a constant sequence)
$\frac{a_2}{a_1} = \frac{2}{1} = 2$ , $\frac{a_3}{a_2} = \frac{3}{2} = 1.5$ , $\frac{a_4}{a_3} = \frac{4}{3} \approx 1.33$

The ratios between adjacent terms are not equal, so it is not a geometric sequence
$\frac{a_2}{a_1} = \frac{8}{16} = \frac{1}{2}$ , $\frac{a_3}{a_2} = \frac{4}{8} = \frac{1}{2}$ , $\frac{a_4}{a_3} = \frac{2}{4} = \frac{1}{2}$

It is a geometric sequence with common ratio $q = \frac{1}{2}$

Exercise 2

Given that the general term of the sequence $\{a_n\}$ is $a_n = 5 \cdot (-2)^{n-1}$ , determine whether the sequence is a geometric sequence. If so, find the common ratio.

Reference Answer(2 个标签)

geometric sequencecommon ratio

Idea: Use the general term method to see whether it can be written in the form $c \cdot q^{n-1}$ .

Detailed steps:

$a_n = 5 \cdot (-2)^{n-1}$

This is of the form $c \cdot q^{n-1}$ with $c = 5, q = -2$ .

Therefore, $\{a_n\}$ is a geometric sequence with common ratio $q = -2$ .

Verification:

$a_1 = 5 \cdot (-2)^0 = 5$
$a_2 = 5 \cdot (-2)^1 = -10$
$a_3 = 5 \cdot (-2)^2 = 20$
$\frac{a_2}{a_1} = \frac{-10}{5} = -2$
$\frac{a_3}{a_2} = \frac{20}{-10} = -2$

Indeed a geometric sequence with common ratio $-2$ .

Exercise 3

In the geometric sequence $\{a_n\}$ , $a_2 = 6$ and $a_5 = 48$ . Find the common ratio $q$ .

Reference Answer(2 个标签)

geometric sequencecommon ratio

Idea: Use the definition of a geometric sequence: the ratio between adjacent terms is the common ratio.

Detailed steps:

From $a_2$ to $a_5$ , the common ratio is applied 3 times:

$a_5 = a_2 \cdot q^3$

Substitute the given values:

$48 = 6 \cdot q^3$ $q^3 = 8$ $q = 2$

Answer: the common ratio is $q = 2$

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$q$	参数	quotient	The common ratio of a geometric sequence
$\{a_n\}$	数列记号	sequence notation	Denotes a sequence
$a_n$	元素符号	a sub n	The $n$ th term of a sequence
$\mathbb{N}^*$	数学符号	positive integers	The set of positive integers

中英对照

中文术语	英文术语	音标	说明
等比数列	geometric sequence	/ˌdʒiːəˈmetrɪk ˈsiːkwəns/	A sequence with a constant ratio between adjacent terms
公比	common ratio	/ˈkɒmən ˈreɪʃiəʊ/	The ratio between adjacent terms of a geometric sequence
指数增长	exponential growth	/ˌekspəˈnenʃəl ɡrəʊθ/	Growing by a fixed multiple
线性增长	linear growth	/ˈlɪniə ɡrəʊθ/	Growing by a fixed amount
常数列	constant sequence	/ˈkɒnstənt ˈsiːkwəns/	A sequence whose terms are all the same

Geometric Sequence

General Term Formula of Geometric Sequences