General Term Formula of Arithmetic Sequences

The general term formula is the core tool of arithmetic sequences. It allows us to directly find any term of the sequence without computing term by term.

Derivation of the Formula

For an arithmetic sequence $\{a_n\}$, let the first term be $a_1$ and the common difference be $d$. Let us derive the expression for the $n$th term $a_n$.

According to the definition of an arithmetic sequence:

$$\begin{aligned} a_2 &= a_1 + d \\ a_3 &= a_2 + d = a_1 + 2d \\ a_4 &= a_3 + d = a_1 + 3d \\ a_5 &= a_4 + d = a_1 + 4d \\ &\vdots \end{aligned}$$

Observe the pattern: from $a_1$ to $a_n$, we need to add $(n-1)$ common differences $d$.

Variations of the General Term Formula

The general term formula can also be written in other forms:

Form 1: Based on an Arbitrary Term

If the $m$th term $a_m$ is known, then:

$a_n = a_m + (n-m)d$

This formula is especially useful when a certain term (not necessarily the first) is known.

Form 2: The Linear Function Form

Expanding the general term formula:

$a_n = a_1 + (n-1)d = dn + (a_1 - d)$

Let $k = d$ and $b = a_1 - d$. Then:

$a_n = kn + b$

This shows that the general term formula of an arithmetic sequence is a linear function of $n$ (when $d \neq 0$).

Worked Examples

Example 1: Given the first term and common difference, find a certain term

In the arithmetic sequence $\{a_n\}$, the first term is $a_1 = 3$ and the common difference is $d = 2$. Find the 10th term $a_{10}$.

Solution:

$a_{10} = a_1 + (10-1)d = 3 + 9 \times 2 = 3 + 18 = 21$

Example 2: Given two terms, find the general term formula

In the arithmetic sequence $\{a_n\}$, $a_3 = 7$ and $a_7 = 15$. Find the general term formula.

Solution:

Using the general term formula:

$$\begin{cases} a_3 = a_1 + 2d = 7 \\ a_7 = a_1 + 6d = 15 \end{cases}$$

Subtracting the two equations: $4d = 8 \quad \Rightarrow \quad d = 2$

Substituting into the first equation: $a_1 + 2 \times 2 = 7 \quad \Rightarrow \quad a_1 = 3$

Therefore, the general term formula is: $a_n = 3 + (n-1) \times 2 = 2n + 1$

Example 3: Determine whether a number is a term of the sequence

In the arithmetic sequence $\{a_n\}$, $a_1 = 5$ and $d = 3$. Is $101$ a term of this sequence? If so, which term?

Solution:

Suppose $101$ is the $n$th term. Then: $a_n = 5 + (n-1) \times 3 = 101$ $3n + 2 = 101$ $3n = 99$ $n = 33$

Since $n = 33$ is a positive integer, $101$ is the 33rd term of this sequence.

Practice Problems

Exercise 1

In the arithmetic sequence $\{a_n\}$, $a_1 = -5$ and $d = 3$. Find $a_{20}$.

Exercise 2

In the arithmetic sequence $\{a_n\}$, $a_4 = 10$ and $a_9 = 25$. Find the first term $a_1$ and the common difference $d$.

Exercise 3

In the arithmetic sequence $\{a_n\}$, $a_1 = 2$ and $d = 5$. Is $97$ a term of this sequence?

Exercise 4

Adapted from a postgraduate entrance examination problem

The arithmetic sequence $\{a_n\}$ satisfies $a_2 + a_4 = 14$ and $a_3 + a_6 = 21$. Find $a_1$ and $d$.

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