Properties of Arithmetic Sequences

Besides the basic definition and formulas, arithmetic sequences have many important properties. Mastering these properties allows you to solve problems more flexibly, and even discover some elegant solutions.

The Middle-Term Property

The Arithmetic Mean

In other words, the arithmetic mean is the arithmetic average of two numbers.

Example: $2$, $5$, $8$ form an arithmetic sequence, and $5$ is the arithmetic mean of $2$ and $8$, because $5 = \frac{2 + 8}{2}$.

The General Middle-Term Property

In the arithmetic sequence $\{a_n\}$, for any positive integer $n$ ($n \geq 2$):

This shows that in an arithmetic sequence, any term is the arithmetic mean of its two neighbors.

The Symmetry Property

In the arithmetic sequence $\{a_n\}$, if $m + n = p + q$ (where $m, n, p, q$ are positive integers), then:

In particular, when $m + n = 2p$, we have $a_m + a_n = 2a_p$.

Example: In an arithmetic sequence, $a_2 + a_8 = a_3 + a_7 = a_4 + a_6 = 2a_5$ (because $2+8=3+7=4+6=2 \times 5=10$)

Proof:

$$\begin{aligned} a_m + a_n &= [a_1 + (m-1)d] + [a_1 + (n-1)d] \\ &= 2a_1 + (m+n-2)d \\ \\ a_p + a_q &= [a_1 + (p-1)d] + [a_1 + (q-1)d] \\ &= 2a_1 + (p+q-2)d \end{aligned}$$

Since $m + n = p + q$, we have $a_m + a_n = a_p + a_q$.

The Subsequence Property

Extracting at Equal Intervals

Extracting several terms from an arithmetic sequence at equal intervals produces a new sequence that is still an arithmetic sequence.

Example: The sequence $\{a_n\}$: $1, 3, 5, 7, 9, 11, 13, \ldots$ (common difference 2)

Extracting odd-indexed terms: $a_1, a_3, a_5, \ldots$ i.e., $1, 5, 9, \ldots$ (common difference 4)

Extracting even-indexed terms: $a_2, a_4, a_6, \ldots$ i.e., $3, 7, 11, \ldots$ (common difference 4)

Sums of Consecutive k Terms

Taking the sum of every consecutive $k$ terms of an arithmetic sequence as a term, the new sequence thus formed is still an arithmetic sequence.

Let $S_k$, $S_{2k} - S_k$, $S_{3k} - S_{2k}$, $\ldots$ form the new sequence. This new sequence is an arithmetic sequence with common difference $k^2d$.

Relationship with the Sum of the First n Terms

The Relationship between the General Term and the Sum of the First n Terms

For an arithmetic sequence $\{a_n\}$, the sum $S_n$ of the first $n$ terms and the general term $a_n$ satisfy:

$$a_n = \begin{cases} S_1, & n = 1 \\ S_n - S_{n-1}, & n \geq 2 \end{cases}$$

The Quadratic Function Property of the Sum

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is a quadratic function of $n$ (when $d \neq 0$):

$S_n = \frac{d}{2}n^2 + \left(a_1 - \frac{d}{2}\right)n$

This is a quadratic function with no constant term.

Properties:

• When $d > 0$, $S_n$ has a minimum
• When $d < 0$, $S_n$ has a maximum
• When $d = 0$, $S_n = na_1$ (a linear function)

Worked Examples

Example 1: Using the Symmetry Property

In the arithmetic sequence $\{a_n\}$, given $a_3 + a_7 = 20$, find $a_5$.

Solution:

Since $3 + 7 = 2 \times 5$, by the symmetry property:

$a_3 + a_7 = 2a_5$

Therefore: $2a_5 = 20 \quad \Rightarrow \quad a_5 = 10$

Example 2: Using the Subsequence Property

The common difference of the arithmetic sequence $\{a_n\}$ is $d = 2$. Extract the 1st, 3rd, 5th, … terms to form a new sequence $\{b_n\}$. Find the common difference of $\{b_n\}$.

Solution:

$b_1 = a_1$, $b_2 = a_3$, $b_3 = a_5$, $\ldots$

$b_2 - b_1 = a_3 - a_1 = 2d = 2 \times 2 = 4$

So the common difference of the new sequence is $4$.

Example 3: Using the Sum Property

The sum of the first $n$ terms of the arithmetic sequence $\{a_n\}$ is $S_n = 2n^2 + 3n$. Find $a_1$ and $d$.

Solution:

When $n = 1$: $a_1 = S_1 = 2 \times 1^2 + 3 \times 1 = 5$

When $n \geq 2$: $a_n = S_n - S_{n-1} = (2n^2 + 3n) - [2(n-1)^2 + 3(n-1)]$ $= 2n^2 + 3n - 2n^2 + 4n - 2 - 3n + 3 = 4n + 1$

Verify $n = 1$: $a_1 = 4 \times 1 + 1 = 5$ ✓

So $a_n = 4n + 1$, i.e., $a_1 = 5$ and $d = 4$.

Practice Problems

Exercise 1

In the arithmetic sequence $\{a_n\}$, $a_4 + a_6 = 18$. Find $a_5$.

Exercise 2

In the arithmetic sequence $\{a_n\}$, $a_2 + a_4 + a_6 + a_8 + a_{10} = 80$. Find $a_6$.

Exercise 3

The sum of the first $n$ terms of the arithmetic sequence $\{a_n\}$ is $S_n = n^2 - n$. Find $a_n$.

Exercise 4

Adapted from a postgraduate entrance examination problem

In the arithmetic sequence $\{a_n\}$, $a_1 > 0$ and $S_9 = S_{12}$. After how many terms does the sum reach its maximum?

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