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Definition of Arithmetic Sequences

The arithmetic sequence is the most basic and important type of sequence. Understanding its definition is the key to mastering all subsequent content.

Definition

Definition of an arithmetic sequence

If, starting from the second term, the difference between each term and its predecessor in a sequence is always the same constant, then the sequence is called an arithmetic sequence. This constant is called the common difference of the arithmetic sequence, usually denoted by the letter $d$ .

符号说明

符号	类型	读音/说明	在本文中的含义
$d$	参数	difference	The common difference of an arithmetic sequence, the difference between adjacent terms

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

$a_{n+1} - a_n = d \quad (n \in \mathbb{N}^*, d \text{ is a constant})$

then $\{a_n\}$ is an arithmetic sequence.

Methods of Determination

There are several methods to determine whether a sequence is an arithmetic sequence:

Method 1: The Definition Method

Check whether the difference between adjacent terms is constant.

Example 1: Determine whether the sequence $2, 5, 8, 11, 14, \ldots$ is an arithmetic sequence.

Solution:

$a_2 - a_1 = 5 - 2 = 3$
$a_3 - a_2 = 8 - 5 = 3$
$a_4 - a_3 = 11 - 8 = 3$
$a_5 - a_4 = 14 - 11 = 3$

The difference between adjacent terms is always 3, so this is an arithmetic sequence with common difference 3.

Method 2: The General Term Method

If the general term formula of a sequence can be written as $a_n = kn + b$ (where $k, b$ are constants), then the sequence is an arithmetic sequence with common difference $d = k$ .

Example 2: Is the sequence $a_n = 3n - 1$ an arithmetic sequence?

Solution: $a_n = 3n - 1$

This is of the form $kn + b$ with $k = 3, b = -1$ , so it is an arithmetic sequence with common difference $d = 3$ .

Verification: $a_1 = 2, a_2 = 5, a_3 = 8, \ldots$ , indeed an arithmetic sequence with common difference 3.

The Meaning of the Common Difference

The common difference $d$ determines the trend of the arithmetic sequence:

$d > 0$ : the sequence is increasing (e.g., $1, 3, 5, 7, \ldots$ )
$d < 0$ : the sequence is decreasing (e.g., $10, 7, 4, 1, \ldots$ )
$d = 0$ : the sequence is constant (e.g., $5, 5, 5, 5, \ldots$ )

Why is a constant sequence (a sequence whose terms are all the same) also an arithmetic sequence?

Because a constant sequence satisfies the definition of an arithmetic sequence: the difference between adjacent terms is a constant. For a constant sequence, every term is the same, so the difference between adjacent terms is 0, i.e., the common difference $d = 0$ . Although it looks like “no change”, from the viewpoint of the mathematical definition, it fully satisfies the conditions of an arithmetic sequence.

Special Cases

A Sequence with Only One Term

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

A Sequence with Only Two Terms

Any two-term sequence is an arithmetic sequence, with common difference $d = a_2 - a_1$ .

Practice Problems

Exercise 1

Determine whether the following sequences are arithmetic sequences. If so, state the common difference:

$1, 4, 7, 10, 13, \ldots$
$5, 5, 5, 5, 5, \ldots$
$1, 2, 4, 8, 16, \ldots$
$-3, -1, 1, 3, 5, \ldots$

Reference Answer(2 个标签)

arithmetic sequencecommon difference

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

Detailed steps:

$a_2 - a_1 = 4 - 1 = 3$ , $a_3 - a_2 = 7 - 4 = 3$ , $a_4 - a_3 = 10 - 7 = 3$

It is an arithmetic sequence with common difference $d = 3$
$a_2 - a_1 = 5 - 5 = 0$ , $a_3 - a_2 = 5 - 5 = 0$

It is an arithmetic sequence with common difference $d = 0$ (a constant sequence)
$a_2 - a_1 = 2 - 1 = 1$ , $a_3 - a_2 = 4 - 2 = 2$ , $a_4 - a_3 = 8 - 4 = 4$

The differences between adjacent terms are not equal, so it is not an arithmetic sequence
$a_2 - a_1 = -1 - (-3) = 2$ , $a_3 - a_2 = 1 - (-1) = 2$ , $a_4 - a_3 = 3 - 1 = 2$

It is an arithmetic sequence with common difference $d = 2$

Exercise 2

Given that the general term of the sequence $\{a_n\}$ is $a_n = 2 - 3n$ , determine whether the sequence is an arithmetic sequence. If so, find the common difference.

Reference Answer(2 个标签)

arithmetic sequencecommon difference

Idea: Use the general term method to see whether it can be written in the form $kn + b$ .

Detailed steps:

$a_n = 2 - 3n = -3n + 2$

This is of the form $kn + b$ with $k = -3, b = 2$ .

Therefore, $\{a_n\}$ is an arithmetic sequence with common difference $d = -3$ .

Verification:

$a_1 = 2 - 3 = -1$
$a_2 = 2 - 6 = -4$
$a_3 = 2 - 9 = -7$
$a_2 - a_1 = -4 - (-1) = -3$
$a_3 - a_2 = -7 - (-4) = -3$

Indeed an arithmetic sequence with common difference $-3$ .

Exercise 3

In the arithmetic sequence $\{a_n\}$ , $a_3 = 7$ and $a_5 = 13$ . Find the common difference $d$ .

Reference Answer(2 个标签)

arithmetic sequencecommon difference

Idea: Use the definition of an arithmetic sequence: the difference between adjacent terms is the common difference.

Detailed steps:

From $a_3$ to $a_5$ there are 2 common differences:

$a_5 - a_3 = 2d$

Substitute the given values:

$13 - 7 = 2d$ $6 = 2d$ $d = 3$

Answer: the common difference is $d = 3$

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$d$	参数	difference	The common difference of an arithmetic 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
等差数列	arithmetic sequence	/ˌærɪθˈmetɪk ˈsiːkwəns/	A sequence with a constant difference between adjacent terms
公差	common difference	/ˈkɒmən ˈdɪfrəns/	The difference between adjacent terms of an arithmetic sequence
递增	increasing	/ɪnˈkriːsɪŋ/	The terms of a sequence gradually get larger
递减	decreasing	/dɪˈkriːsɪŋ/	The terms of a sequence gradually get smaller
常数列	constant sequence	/ˈkɒnstənt ˈsiːkwəns/	A sequence whose terms are all the same

Arithmetic Sequence

General Term Formula of Arithmetic Sequences