A-Level Mathematics: Sequences and Series — An Overview | A-Level 数学：数列与级数概述

📚 A-Level Mathematics: Sequences and Series — An Overview | A-Level 数学：数列与级数概述

Sequences and series are central topics in the CIE A-Level Mathematics syllabus. They appear in Paper 1 (Pure Mathematics 1) and are extended in Pure Mathematics 2/3. This article gives an exam-focused overview: key definitions, formulas, worked examples, and common traps that students must avoid.

数列与级数是 CIE A-Level 数学考试的核心内容，出现在 Paper 1（Pure Mathematics 1），并在 Pure Mathematics 2/3 中进一步加深。本文以考试为导向，为你系统梳理核心定义、公式、典型例题以及常见易错点。

1. Sequences and Series Explained | 数列与级数的定义

A sequence is a list of numbers written in a definite order, such as 2, 4, 6, 8, … . Each number in the list is called a term. The terms are usually denoted by u₁, u₂, u₃, …, and the n-th term can often be written as a formula involving n.

数列是按确定顺序排列的一列数，例如 2, 4, 6, 8, …。其中的每一个数称为“项”，通常记作 u₁, u₂, u₃, …。许多数列的第 n 项都可以写成关于 n 的公式。

A series is what you obtain when you add the terms of a sequence together. For example, if the sequence is 1, 2, 3, 4, then the corresponding series is 1 + 2 + 3 + 4 = 10.

级数（series）是指把数列中的各项依次相加所得到的结果。例如，若数列是 1, 2, 3, 4，则对应的级数为 1 + 2 + 3 + 4 = 10。

A finite sequence has a last term, while an infinite sequence continues without end. The same distinction applies to finite and infinite series.

有限数列有最后一项，而无穷数列则没有尽头。相应地，级数也分为“有限级数”和“无穷级数”。

2. Arithmetic Sequences | 等差数列

An arithmetic sequence, often called an arithmetic progression (AP), is a sequence in which the difference between consecutive terms is constant. This constant is known as the common difference and is usually denoted by d.

等差数列（通常记为 AP）是指相邻两项之差为常数的数列。这个常数称为“公差”，通常用 d 表示。

If the first term is a, then the terms are a, a + d, a + 2d, a + 3d, and so on. The n-th term formula is central to this topic.

若首项为 a，则数列为 a, a + d, a + 2d, a + 3d, …。其中“第 n 项公式”是这个知识点的核心。

uₙ = a + (n − 1)d

Notice that the coefficient of d is n − 1, not n. This is one of the most common errors in sequences questions.

注意公差前面的系数是 n − 1，而不是 n。这是数列题中最常见的错误之一。

For example, for the AP 3, 7, 11, 15, … , first term a = 3 and common difference d = 4. The 20th term is u₂₀ = 3 + 19 × 4 = 79.

例如，在等差数列 3, 7, 11, 15, … 中，首项 a = 3，公差 d = 4。第 20 项为 u₂₀ = 3 + 19 × 4 = 79。

3. Geometric Sequences | 等比数列

A geometric sequence, or geometric progression (GP), is a sequence in which the ratio between consecutive terms is constant. This constant ratio is denoted by r.

等比数列（通常记为 GP）是指相邻两项之比为常数的数列。这个常数称为“公比”，通常用 r 表示。

If the first term is a, then the terms are a, ar, ar², ar³, and so on. The n-th term is given by the formula below.

若首项为 a，则数列为 a, ar, ar², ar³, …。它的第 n 项公式如下。

uₙ = arⁿ⁻¹

Again, the power is n − 1, because the first term a is ar⁰. For example, in the GP 2, 6, 18, 54, … , a = 2 and r = 3. The 6th term is 2 × 3⁵ = 486.

再一次提醒：指数是 n − 1，因为首项 a 可以看作 ar⁰。例如在等比数列 2, 6, 18, 54, … 中，a = 2，r = 3。第 6 项为 2 × 3⁵ = 486。

If the ratio r is positive, the terms have a consistent sign. If r is negative, the signs alternate. If |r| = 1, the sequence is constant or simply alternates between two values.

若公比 r 为正，数列各项符号一致；若 r 为负，数列正负交替；若 |r| = 1，数列为常数或仅在两个值之间交替。

4. Summing an Arithmetic Series | 等差级数求和

The sum of the first n terms of an arithmetic progression is denoted by S

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