📚 Arithmetic Sequences: General Term Formula and Properties | 等差数列通项公式与性质
Arithmetic sequences are one of the most fundamental topics in algebra and appear frequently in A-Level Mathematics examinations. Understanding the general term formula and the key properties of arithmetic sequences is essential for solving problems efficiently and accurately.
等差数列是代数中最基础的内容之一，也是 A-Level 数学考试中的高频考点。掌握等差数列的通项公式及其核心性质，是高效、准确解题的关键。
1. Definition and Notation | 定义与记号
An arithmetic sequence, also called an arithmetic progression, is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is known as the common difference and is usually denoted by d.
等差数列（又称算术级数）是指任意两个相邻项之差恒为常数的数列。这个常数称为公差，通常用 d 表示。
For example, 2, 5, 8, 11, 14 is an arithmetic sequence with first term a₁ = 2 and common difference d = 3, since 5 − 2 = 8 − 5 = 11 − 8 = 14 − 11 = 3.
例如，2, 5, 8, 11, 14 是一个等差数列，首项 a₁ = 2，公差 d = 3，因为 5 − 2 = 8 − 5 = 11 − 8 = 14 − 11 = 3。
Standard notation: a₁ denotes the first term, aₙ denotes the n-th term, and n is a positive integer. The common difference is defined as d = aₙ₊₁ − aₙ for all valid n.
标准记号：a₁ 表示首项，aₙ 表示第 n 项，n 为正整数。公差定义为 d = aₙ₊₁ − aₙ，对所有有效的 n 恒成立。
2. The General Term Formula | 通项公式
The n-th term of an arithmetic sequence is given by the following general term formula:
等差数列第 n 项的通项公式如下：
aₙ = a₁ + (n − 1)d
This formula allows us to compute any term directly if we know the first term, the common difference, and the position n.
只要知道首项、公差和项的位置 n，就可以直接计算出任意一项。
For example, given a₁ = 3 and d = 4, the 10th term is a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39.
例如，已知 a₁ = 3，d = 4，则第 10 项为 a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39。
The formula can also be rewritten as aₙ = dn + (a₁ − d), which reveals that the n-th term is a linear function of n with slope d. This perspective is useful when analysing the behaviour of the sequence.
通项公式还可改写为 aₙ = dn + (a₁ − d)，这表明第 n 项是 n 的一次函数，斜率为 d。这一视角有助于分析数列的变化规律。
3. Extended General Term Formula | 延伸通项公式
When the first term is unknown but another term such as the m-th term is given, we can use the extended general term formula:
当首项未知，但已知某一项（如第 m 项）时，可使用延伸通项公式：
aₙ = aₘ + (n − m)d
This formula is particularly convenient when the given information centres on a term other than the first.
当题目给出的信息围绕非首项的某一项时，这一公式尤为便利。
For example, if a₅ = 20 and d = 3, then a₁₀ = a₅ + (10 − 5)d = 20 + 5 × 3 = 35.
例如，若 a₅ = 20，d = 3，则 a₁₀ = a₅ + (10 − 5)d = 20 + 5 × 3 = 35。
This formula also allows us to recover the first term: a₁ = aₘ − (m − 1)d. It is especially useful when solving systems of equations involving two unknown terms.
利用该公式还可反推首项：a₁ = aₘ − (m − 1)d。当需要通过方程组求解两个未知项时，这个公式尤为有用。
4. Key Properties of Arithmetic Sequences | 等差数列的主要性质
Arithmetic sequences possess several important properties that are frequently tested in examinations.
等差数列具有若干重要性质，这些性质在考试中经常出现。
Property 1: If m + n = p + q, then aₘ + aₙ = aₚ + a_q.
性质 1：若 m + n = p + q，则 aₘ + aₙ = aₚ + a_q。
This property reflects the symmetry of arithmetic sequences: the sum of any two terms depends only on the sum of their index positions.
该性质体现了等差数列的对称性：任意两项之和仅取决于它们的下标之和。
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For example: if a₂ + a₈ = 30, then a₄ + a₆ = 30 as well, because 2 + 8 = 4 + 6 = 10.
例如：若 a₂ + a₈ = 30，则 a₄ + a₆ = 30，因为 2 +
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