Finding a Formula for a Sequence | 数列公式求解指南

📚 Finding a Formula for a Sequence | 数列公式求解指南

Sequences appear throughout the Edexcel IGCSE Mathematics syllabus, from simple number patterns to the algebra of quadratic expressions. A sequence is an ordered list of numbers, and the key skill is to find a formula that gives any term directly from its position. This revision guide breaks down the process for linear, geometric and quadratic sequences, with worked examples and exam-style tips.

在 Edexcel IGCSE 数学考纲中，数列贯穿始终，从简单的数字模式到二次表达式的代数运算都会涉及。数列是按顺序排列的一列数，而核心技能是找到一条公式，直接从项的位置得出任意一项。本复习指南将逐步讲解线性数列、等比数列和二次数列的求解方法，并配有例题与考试技巧。

1. What is a Sequence? | 什么是数列

A sequence is a list of numbers written in a definite order, such as 2, 5, 8, 11, … Each number in the list is called a term. The first term is written u₁, the second term is written u₂, and the term in position n is written uₙ. You can describe the whole list either by giving the first few terms or by giving a formula for uₙ.

数列是按一定顺序排列的一列数，例如 2, 5, 8, 11, ……。列表中的每一个数称为一项。第一项记作 u₁，第二项记作 u₂，第 n 项记作 uₙ。你可以通过给出前几项或给出 uₙ 的公式来描述整个数列。

There are two types of rule. A term-to-term rule tells you how to move from one term to the next, such as “add 3”. A position-to-term rule, often called the nth-term formula, tells you how to find a term directly from its position n. The exam usually asks for the position-to-term rule, because it can find any term without listing all the previous ones.

规则分为两类。递推规则（项到项的规则）告诉你如何从一项得到下一项，例如”每次加 3″。而通项规则（第 n 项公式）告诉你如何直接从位置 n 求出某一项。考试通常要求的是第 n 项公式，因为有了它就可以不逐一列举前面的项而直接求出任意一项。

2. Key Notation: uₙ and n | 关键记号：uₙ 与 n

The symbol uₙ represents the nth term, where n is a positive integer 1, 2, 3, … To generate a sequence from a formula, substitute n = 1, 2, 3 in turn. For example, if uₙ = 2n + 3, then u₁ = 2(1) + 3 = 5, u₂ = 2(2) + 3 = 7, and u₃ = 2(3) + 3 = 9, so the sequence begins 5, 7, 9, …

记号 uₙ 表示第 n 项，其中 n 是正整数 1, 2, 3, ……。要从公式生成数列，只需依次代入 n = 1, 2, 3。例如，若 uₙ = 2n + 3，则 u₁ = 2(1) + 3 = 5，u₂ = 2(2) + 3 = 7，u₃ = 2(3) + 3 = 9，因此该数列开头为 5, 7, 9, ……。

A common exam skill is the reverse process: given a sequence, find uₙ. Throughout this guide, always test your formula by substituting n = 1 and n = 2 to confirm it reproduces the first two given terms exactly.

考试中常见的逆向技能是：已知数列，求 uₙ。在整个复习过程中，务必通过代入 n = 1 和 n = 2 检验你的公式，确认它能准确还原题目给出的前两项。

3. Arithmetic Sequences | 等差数列

An arithmetic sequence has a constant difference d between consecutive terms. For example, 4, 7, 10, 13, … has d = 3, because 7 – 4 = 3 and 10 – 7 = 3. The word “arithmetic” here comes from the idea of adding or subtracting the same number each time.

等差数列相邻两项的差 d 保持不变。例如 4, 7, 10, 13, ……

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