📚 Sequences and Series for CIE GCSE Mathematics | 数列与级数考点精讲
This article provides a comprehensive revision guide for sequences and series in the CIE GCSE Mathematics syllabus. It covers arithmetic and geometric progressions, nth term formulas, summation techniques, infinite geometric series, recurrence relations, sigma notation and real-world applications. Understanding these core topics is essential for exam success.
本文为 CIE GCSE 数学课程中的数列与级数提供全面复习指南，涵盖等差数列、等比数列、通项公式、求和方法、无穷几何级数、递推关系、求和符号与实际应用。掌握这些核心考点对考试至关重要。
1. Introduction to Sequences | 数列简介
a term, and every term has a position, usually denoted by the letter n (with n = 1, 2, 3, …).
数列是按特定规则或模式排列的一列有序数字。列表中的每个数字称为项，每一项都有位置，通常用字母 n 表示（n = 1, 2, 3, …）。
For example, the sequence 4, 7, 10, 13, … follows the rule 'add 3 to the previous term'. This is called a recurrence relation. We can also describe a sequence using an explicit formula for the nth term, which allows us to find any term directly without knowing the previous ones.
例如，数列 4, 7, 10, 13, … 遵循规则“前一项加 3”。这称为递推关系。我们也可以使用通项公式来描述数列，从而在不需要知道前一项的情况下直接求出任意项。
A finite sequence has a fixed number of terms (e.g. the first five multiples of 3: 3, 6, 9, 12, 15). An infinite sequence continues without end (e.g. the natural numbers 1, 2, 3, …).
有限数列具有固定的项数（例如 3 的前五个倍数：3, 6, 9, 12, 15）。无限数列则无限延续（例如自然数 1, 2, 3, …）。
2. Arithmetic Sequences (AP) | 等差数列
In an arithmetic sequence, the difference between any term and the preceding term is constant. This constant difference is called the common difference, usually denoted by d.
在等差数列中，任意一项与其前一项的差是常数。这个常数差称为公差，通常记为 d。
If the first term is a, the sequence can be written as: a, a + d, a + 2d, a + 3d, … .
如果首项是 a，数列可写为：a, a + d, a + 2d, a + 3d, … 。
Example: 8, 13, 18, 23, … has a = 8 and d = 5. Another example: 20, 14, 8, 2, … arithmetic sequence, subtract consecutive terms and see if the differences are all equal.
例子：8, 13, 18, 23, … 的 a = 8，d = 5。另一个例子：20, 14, 8, 2, … 的 a = 20，d = -6。要判断一组数字是否为等差数列，可计算相邻项的差是否全都相等。
3. The nth Term of an Arithmetic Sequence | 等差数列的通项公式
The nth term (often written as uₙ or aₙ) of an arithmetic sequence is obtained by starting with the first term a and adding the common difference d repeatedly (n – 1) times. This gives the formula:
等差数列的第 n 项（通常写作 uₙ 或 aₙ）是通过从首项 a 开始，重复加上公差 d 共 (n – 1) 次得到。公式如下：
aₙ = a + (n – 1)d
For example, find the 20th term of the sequence 3, 8, 13, 18, … . Here a = 3, d = 5, n = 20. Substituting: a₂₀ = 3 + (20 – 1) × 5 = 3 + 95 = 98.
例如，求数列 3, 8, 13, 18, … 的第 20 项。这里 a = 3，d = 5，n = 20。代入得 a₂₀ = 3 + (20 – 1) × 5 = 3 + 95 = 98。
we can solve for n. In an arithmetic sequence with a = 10 and d = 3, which term equals 40? Set up 10 + (n – 1) × 3 = 40 → (n – 1) × 3 = 30 → n – 1 = 10 → n = 11. So the 11th term is 40.
如果已知
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