Binomial Expansion: Extracting Unknowns from Given Coefficients (Edexcel A-Level) | 二项式展开：由给定系数反求未知数（爱德思A-Level）

📚 Binomial Expansion: Extracting Unknowns from Given Coefficients (Edexcel A-Level) | 二项式展开：由给定系数反求未知数（爱德思A-Level）

In Edexcel A‑Level Mathematics, binomial expansion questions often go beyond simple expansion. You may encounter problems where a particular coefficient is given—such as 228—and you must determine the unknown power n or a constant k. This article will equip you with a systematic approach to tackle these “reverse” problems, from setting up the general term to solving the resulting equations, and will highlight common pitfalls and exam strategies.

在爱德思A‑Level数学中，二项式展开题常常不只要求直接展开。你可能会遇到这样的问题：题目给出了某一项的具体系数——例如228——然后要求你求出未知的指数n或某个常数k。本文将为你提供一套系统的方法来应对这类“逆推”题，包括建立通项、求解得到的方程，并着重指出常见错误和应试策略。

1. Binomial Theorem – The Foundation | 二项式定理——基础回顾

The binomial theorem states that for a positive integer n, the expansion of (a + b)ⁿ is given by a sum of terms involving binomial coefficients. In Edexcel notation, the coefficient of a^n−rb^r is commonly written as ⁿC_r or . Mastering this form is essential before you can handle reverse problems.

二项式定理指出，对于正整数n，(a + b)ⁿ 的展开式由一系列包含二项式系数的项相加而成。在爱德思考试中，a^n−rb^r 的系数通常写作 ⁿC_r。在应对逆推问题之前，必须熟练掌握这一形式。

(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + ⁿC_n bⁿ

(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + ⁿC_n bⁿ

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