Binomial Expansion: CIE A-Level Maths Key Points | CIE A-Level 数学二项式展开考点精讲

📚 Binomial Expansion: CIE A-Level Maths Key Points | CIE A-Level 数学二项式展开考点精讲

The binomial expansion is a fundamental topic in CIE A-Level Mathematics (both Pure 1 and Pure 3), appearing frequently in algebraic manipulation, series approximations, and real-world modelling. Mastering the binomial theorem for positive integer powers and the extension to rational indices is essential for scoring highly on Paper 1 and Paper 3.

二项式展开是 CIE A-Level 数学（包括纯数 1 和纯数 3）的基础主题，经常出现于代数化简、级数近似和实际建模中。掌握正整数幂的二项式定理以及向有理指数推广，对于在试卷 1 和试卷 3 取得高分至关重要。

1. Binomial Theorem for Positive Integer Powers | 正整数幂的二项式定理

For a positive integer n, the expansion of (a + b)ⁿ is given by the sum: (a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + ⁿCᵣ aⁿ⁻ʳ bʳ + … + ⁿCₙ bⁿ, where the binomial coefficient ⁿCᵣ = n! / (r!(n–r)!).

对于正整数 n，(a + b)ⁿ 的展开由以下求和给出：(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + ⁿCᵣ aⁿ⁻ʳ bʳ + … + ⁿCₙ bⁿ，其中二项式系数 ⁿCᵣ = n! / (r!(n–r)!)。

(a + b)ⁿ = Σᵣ₌₀ⁿ ⁿCᵣ aⁿ⁻ʳ bʳ

Note that the coefficients are symmetric: ⁿC

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