A-Level Maths: Sigma Summation Notation | A-Level 数学：西格玛求和记号

📚 A-Level Maths: Sigma Summation Notation | A-Level 数学：西格玛求和记号

Sigma notation is one of the most compact and powerful tools in A-Level Mathematics. It allows you to write a long sum as a single expression, to manipulate it algebraically, and to evaluate it quickly using standard results. This article covers everything you need: the definition, the algebraic properties, the three standard sum formulas, their links to arithmetic and geometric series, and step-by-step exam-style examples.

西格玛（Σ）记法是 A-Level 数学中最简洁而强大的工具之一。它让你可以把一个很长的求和写成单个表达式，进行代数化简，并借助标准公式快速求值。本文将涵盖你所需的一切：定义、代数性质、三个标准求和公式、与等差级数和等比级数的联系，以及逐步讲解的考试型例题。

1. What Is Sigma Notation? | 什么是西格玛记号

The capital Greek letter Σ (sigma) stands for ‘sum’. The expression Σ_{r = 1}ⁿ a_r tells you to start at r = 1, calculate the term a_r, and then add up all the terms up to r = n.

大写希腊字母 Σ（西格玛）表示”求和”。表达式 Σ_{r = 1}ⁿ a_r 的意思是：从 r = 1 开始，算出每一项 a_r，然后把直到 r = n 的所有项相加。

For example, Σ_{r = 1}⁵ r² = 1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55.

例如，Σ_{r = 1}⁵ r² = 1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55。

The letter r is called the summation index or dummy variable. It can be replaced by any other letter, so Σ_{r = 1}ⁿ a_r and Σ_{k = 1}ⁿ a_k mean exactly the same thing. This is useful when you need to avoid confusion with another variable already named r in the question.

字母 r 被称为求和指标或哑变量。它可以用任何其他字母替换，因此 Σ_{r = 1}ⁿ a_r 与 Σ<

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