📚 Sequences Fundamentals: Definitions, Classification and Common Question Types | 数列基础精讲：定义、分类与常见题型
In mathematics, a sequence is a list of numbers arranged in a special order. Each number in the list is called a term, and the position of a term is numbered by positive integers 1, 2, 3, ... Sequences are a core topic in the Edexcel IGCSE Mathematics syllabus because they test pattern recognition, algebraic manipulation and problem solving.
在数学中，数列是按照一定顺序排列的一列数。其中的每一个数称为一项，项的位置用正整数 1、2、3…… 编号。数列是 Edexcel IGCSE 数学考纲中的核心内容，因为它同时考查模式识别、代数运算与问题解决能力。
1. Definition and Notation | 数列的定义与记号
A sequence is an ordered list of numbers that usually follows a rule. For example, 2, 4, 6, 8, ... is a sequence. The first term is often written as u₁, the second term as u₂, and the nth term as uₙ.
数列是按顺序排列的一列数，通常遵循某种规律。例如 2、4、6、8…… 就是一个数列。第 1 项通常记作 u₁，第 2 项记作 u₂，第 n 项记作 uₙ。
A rule can be a term-to-term rule, which tells you how to get from one term to the next, or a position-to-term rule, which tells you how to find the nth term directly.
规律可以表现为“项到项规则”，即告诉你如何从一项得到下一项；也可以表现为“位置到项规则”，即直接求出第 n 项。
2. Classification of Sequences | 数列的分类
In IGCSE, you should be able to identify the main types of sequences from their patterns.
在 IGCSE 中，你应该能够根据数列的规律识别其主要类型。
Arithmetic sequence: the difference between consecutive terms is constant.
等差数列：相邻两项的差保持不变。
Geometric sequence: the ratio between consecutive terms is constant.
等比数列：相邻两项的比值保持不变。
Quadratic sequence: the second differences are constant.
二次数列：相邻两项之差再次作差后，得到的二阶差为常数。
Recurrence sequence: each term is defined from one or more previous terms.
递推数列：每一项都由前面的一项或几项定义。
Sequences can also be finite or infinite. A finite sequence has a last term, while an infinite sequence continues forever.
数列还可分为有限数列与无限数列。有限数列有最后一项，无限数列则一直延续下去。
3. Arithmetic Sequences | 等差数列
An arithmetic sequence has a constant difference between every two consecutive terms. This constant is called the common difference and is usually denoted by d.
等差数列相邻两项的差为常数，这个常数称为公差，通常用 d 表示。
is u₁ = 6. The common difference is d = 4 because 10 − 6 = 4, 14 − 10 = 4 and 18 − 14 = 4.
例如，在数列 6、10、14、18 中，首项 u₁ = 6。公差 d = 4，因为 10 − 6 = 4，14 − 10 = 4，18 − 14 = 4。
To check whether a sequence is arithmetic, subtract consecutive terms. If all the results are the same, the sequence is arithmetic.
判断一个数列是否为等差数列，可以逐项相减。如果所有差值都相同，那么它就是等差数列。
4. nth Term of an Arithmetic Sequence | 等差数列的通项
For an arithmetic sequence, the nth term can be written as
uₙ = u₁ + (n − 1)d
This formula works because you start with u₁ and add d each time you move to the next position.
这个公式成立的原因是：你从 u₁ 出发，每向前移动一个位置就加上一个 d。
Example: find the 20th term of the sequence 5, 9, 13, 17, ...
例：求数列 5、9、13、17…… 的第 20 项。
Here u₁ = 5 and d = 4, so u₂₀ = 5 + (20 − 1) × 4 = 5 + 76 = 81.
这里 u₁ = 5，d = 4，所以 u₂₀ = 5 + (20 − 1) × 4 = 5 + 76 = 81。
5. Geometric Sequences | 等比数列
A geometric sequence has a constant ratio between consecutive terms. This constant is called the common ratio and is usually denoted by r.
等比数列相邻两项的比值是常数，这个常数称为公比，通常用 r 表示。
For example, in the sequence 3, 6, 12, 24, the first term
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