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

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., $5, 5, 5, 5, \ldots$)

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

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.

Exercise 3

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

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