Writing a Mathematical Proof: Structure, Framework, and Model Essay for WJEC Further Mathematics

📚 Writing a Mathematical Proof: Structure, Framework, and Model Essay for WJEC Further Mathematics | WJEC进阶数学论文写作：框架与范文

In Year 13 WJEC Further Mathematics, assessment often includes extended questions that require not just a sequence of equations but a coherent, logically structured argument. These ‘mini-essays’ test your ability to communicate mathematical reasoning with precision, explain each deductive step, and present a complete proof. Mastering this format is essential for accessing the top marks on Units such as Further Pure Mathematics A and B, where quality of written communication is explicitly credited. This article provides a step-by-step framework for constructing such proofs, along with a fully worked model essay on de Moivre’s theorem to illustrate best practice.

在 Year 13 WJEC 进阶数学中，考试常包含需要展示连贯逻辑论证的扩展题型，而不仅仅是列出一串等式。这类’小论文’考查你用精确的语言传达数学推理、解释每个演绎步骤并呈现完整证明的能力。掌握这种格式对于在进阶纯数 A 和 B 等单元中冲击最高分至关重要，因为书面表达质量会直接计入评分。本文提供了一个构建此类证明的逐步框架，并配有一篇关于棣莫弗定理的完整范文，以展示最佳实践。

1. Understanding the Requirement of Extended Writing in WJEC Further Maths | 理解 WJEC 进阶数学中长篇写作的要求

WJEC mark schemes reward answers that demonstrate a clear, logical flow and the correct use of mathematical language. An extended proof is not a list of disconnected calculations; it must read as a persuasive argument, where each assertion follows from the previous one. Examiners look for a stated proposition, a well-defined method (e.g. induction, contradiction), and a conclusion that echoes the original statement.

WJEC 评分标准奖励展现出清晰逻辑流程和正确数学语言运用的答案。长篇证明不是一系列脱节的计算，而必须像一篇有说服力的论证，其中每一个断言都严格承接上一个。考官看重的是明确定义命题、清晰选择方法（如归纳法、反证法），以及能呼应原命题的结论。

2. The Core Elements of a Coherent Mathematical Proof | 连贯数学证明的核心要素

Every high-scoring proof contains three distinct sections: an introduction, a body, and a conclusion. The introduction names the theorem or statement to be proved and specifies the domain (e.g. n ∈ ℕ). The body develops the argument step by step, using established axioms or previously proved results. The conclusion restates the result, explicitly confirming that the proof is complete. A common error is omitting the final declarative sentence, which costs a communication mark.

任何高分证明都包含三个泾渭分明的部分：引言、主体和结论。引言指明所要证明的定理或命题，并限定变量范围（例如 n ∈ ℕ）。主体部分一步步展开论证，使用公认的公理或此前已经证明过的结论。结论则重申结果，明确宣告证明已经完成。常见错误是遗漏最后那句宣告性的语句，导致损失表达分。

3. Preparing Your Essay: Planning and Notation | 准备论文：规划与符号

Before writing, jot down a skeleton: state P(n), the base case, the inductive hypothesis, and the inductive step. Choose notation that is unambiguous and consistent; for complex numbers, use z = x + iy or exponential form as appropriate. If the question involves trigonometric identities, decide whether you will use compound-angle formulae or Euler’s relation. A small investment in planning prevents rambling and keeps the proof within a sensible length.

动笔之前，先草拟一个骨架：写出 P(n)、基础情形、归纳假设和归纳步骤。选择明确且前后一致的符号；与复数相关时，按需使用 z = x + iy 或指数形式。如果题目涉及三角恒等式，事先决定要用复合角公式还是欧拉公式。花少量时间规划能防止跑题，并使证明长度适中。

4. Introduction: State the Proposition Clearly | 引言：清晰陈述命题

Open your proof by writing ‘We wish to prove that …’ or ‘Let P(n) be the statement …’. Include all necessary conditions, such as ‘for all positive integers n’ or ‘for all real θ’. For example, when proving de Moivre’s theorem, write:

以’我们欲证明……’或’设 P(n) 为命题……’开头。包含所有必要条件，例如’对所有正整数 n’或’对所有实数 θ’。例如，在证明棣莫弗定理时，可写：

Let P(n): (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, for n ∈ ℕ.

This precise declaration sets the scope and provides a reference point for the concluding line.

这一精确的声明界定了范围，并为最后结论提供了呼应点。

5. Main Body: Step-by-Step Logical Flow | 主体：逐步逻辑流程

The body must be a chain of deductions. If you are using induction, start with the base case: n = 1. Verify that the left-hand side equals the right-hand side. Then state the inductive hypothesis: ‘Assume P(k) is true for some arbitrary k ∈ ℕ.’ Next, consider P(k + 1) and manipulate the expression so that the hypothesis can be applied. Every algebraic manipulation, such as expanding brackets or applying a trig identity, should be accompanied by a brief justification.

主体必须是一条推理链条。若用归纳法，可从基础情形 n = 1 开始。验证左式等于右式。然后陈述归纳假设：’假设对某个任意 k ∈ ℕ，P(k) 成立。’接着考虑 P(k + 1) 并变形，使得能够代用归纳假设。每一次代数操作，比如展开括号或使用三角恒等式，都应附上简短的理由说明。

6. Using Induction: A Powerful Framework | 使用归纳法：一种强大的框架

Mathematical induction is the most common structured-proof format in WJEC Further Pure. The template is:

Base: Show true for n = 1.
Hypothesis: Assume true for n = k.
Step: Prove true for n = k + 1, using the hypothesis.
Conclusion: Hence, by induction, true for all n ∈ ℕ.

This framework transforms a potentially messy proof into a clean, exam-ready argument. Practise it with series, divisibility, and complex numbers.

数学归纳法是 WJEC 进阶纯数中最常见的结构化证明格式。模板为：

基础情形：证明 n = 1 时成立。
假设：假设 n = k 时成立。
步骤：利用假设证明 n = k + 1 时成立。
结论：因此，由归纳法，对所有 n ∈ ℕ 成立。

该框架能将可能混乱的证明转化为清晰、合于考试要求的论证。建议你通过级数、整除性和复数题目勤加练习。

7. Handling Complex Numbers and Trigonometric Forms | 处理复数与三角形式

When proving de Moivre’s theorem, you will multiply two complex numbers in polar form. Recall that the product of (cos A + i sin A) and (cos B + i sin B) equals cos(A + B) + i sin(A + B). In the inductive step, you multiply (cos kθ + i sin kθ) by (cos θ + i sin θ) and use the compound-angle identities:

证明棣莫弗定理时，需要将两个极坐标形式的复数相乘。回顾 (cos A + i sin A) 与 (cos B + i sin B) 的乘积等于 cos(A + B) + i sin(A + B)。在归纳步骤中，你将 (cos kθ + i sin kθ) 乘以 (cos θ + i sin θ)，并使用复合角恒等式：

cos kθ cos θ − sin kθ sin θ = cos(kθ + θ) = cos(k + 1)θ
sin kθ cos θ + cos kθ sin θ = sin(kθ + θ) = sin(k + 1)θ

Mastery of these identities is essential, as they frequently appear in Further Pure contexts.

掌握这些恒等式至关重要，因为它们频繁出现于进阶纯数中。

8. Model Essay: Proving de Moivre’s Theorem by Induction | 范文：用归纳法证明棣莫弗定理

Below is a model answer written in the style expected by WJEC examiners for a high-mark proof question. Read through it, noting how the structure, notation, and commentary align with the framework discussed.

下文是一篇按 WJEC 考官所期望的高分证明题风格写成的范文。请通读并注意其结构、符号与解说如何与前述框架紧密契合。

Proof: We prove by induction that (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for all positive integers n.

证明：我们用归纳法证明对所有正整数 n，有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。

Let P(n) be the statement (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, n ∈ ℕ.

设 P(n) 为命题 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ，n ∈ ℕ。

Base case (n = 1): LHS = (cos θ + i sin θ)¹ = cos θ + i sin θ. RHS = cos(1·θ) + i sin(1·θ) = cos θ + i sin θ. Hence P(1) is true.

基础情形 (n = 1)：左式 = (cos θ + i sin θ)¹ = cos θ + i sin θ。右式 = cos(1·θ) + i sin(1·θ) = cos θ + i sin θ。因此 P(1) 成立。

Inductive hypothesis: Assume P(k) is true for some arbitrary k ∈ ℕ, i.e. (cos θ + i sin θ)^k = cos kθ + i sin kθ.

归纳假设：假设对某个任意 k ∈ ℕ，P(k) 成立，即 (cos θ + i sin θ)^k = cos kθ + i sin kθ。

Inductive step: Consider P(k + 1):

归纳步骤：考虑 P(k + 1)：

(cos θ + i sin θ)^(k + 1) = (cos θ + i sin θ)^k · (cos θ + i sin θ)

Using the inductive hypothesis, this becomes:

利用归纳假设，上式化为：

= (cos kθ + i sin kθ)(cos θ + i sin θ)

Expand the product:

展开乘积：

= cos kθ cos θ + i cos kθ sin θ + i sin kθ cos θ + i² sin kθ sin θ

Since i² = −1, we group real and imaginary parts:

由于 i² = −1，我们将实部与虚部分组：

Real part: cos kθ cos θ − sin kθ sin θ
Imaginary part: i(sin kθ cos θ + cos kθ sin θ)

Apply the compound-angle formulas:

应用复合角公式：

cos kθ cos θ − sin kθ sin θ = cos(kθ + θ) = cos(k + 1)θ
sin kθ cos θ + cos kθ sin θ = sin(kθ + θ) = sin(k + 1)θ

Hence, the whole expression simplifies to:

因此，整个式子化简为：

= cos(k + 1)θ + i sin(k + 1)θ

This is exactly P(k + 1). Thus, if P(k) is true, then P(k + 1) is true.

这正是 P(k + 1)。因此，如果 P(k) 成立，则 P(k + 1) 成立。

Conclusion: Since P(1) is true and P(k) ⇒ P(k + 1), by the principle of mathematical induction, P(n) is true for all n ∈ ℕ. ∎

结论：由于 P(1) 成立且 P(k) ⇒ P(k + 1)，根据数学归纳法原理，对所有 n ∈ ℕ，P(n) 成立。∎

This model answer exemplifies clarity, rigour, and the effective use of the induction framework.

该范文体现了清晰性、严谨性以及对归纳框架的有效运用。

9. Conclusion and Final Checks | 结论与最终检查

Always end with a statement that mirrors the original proposition. The words ‘Hence, by induction, the statement holds for all positive integers n’ are worth a mark. Check that you have not accidentally used the result you are trying to prove within the proof (circular reasoning). Re-read your proof as if you were an examiner: does the logic flow without gaps?

务必以一句呼应原命题的话收尾。’因此，由归纳法，该命题对所有正整数 n 成立’这句话值一分。检查你是否在证明中误用了正要证明的结论（循环论证）。以考官的心态重读你的证明：逻辑是否环环相扣，毫无漏洞？

10. Common Pitfalls and How to Avoid Them | 常见陷阱及如何避免

One frequent mistake is skipping the base case or assuming n = 0 without justification. Another is working backwards: manipulating the target expression until a truth is reached is not a valid proof unless you explicitly state that the steps are reversible. Also, avoid vague phrases like ‘it is obvious that’; instead, cite the specific theorem or identity you are applying. Finally, take care with indices: (cos θ + i sin θ)^k · (cos θ + i sin θ) is straightforward, but students sometimes misplace brackets.

一个常见错误是跳过基础情形，或在无理由的情况下假设 n = 0。另一个是反向推理：一直变形目标表达式直到得出一个真命题，这不构成有效证明，除非你明确声明步骤是可逆的。还要避免使用’显然’这类含糊字眼，而应指明你所用的具体定理或恒等式。最后，注意指数：

(cos θ + i sin θ)^k · (cos θ + i sin θ)

很简单，但学生有时会放错括号。

11. Exam Tips for Maximum Marks | 考试中夺取高分的技巧

Allocate about 10-12 minutes for a full induction proof in the exam. Write neatly and use separate lines for each key equation. Label ‘Base’, ‘Hypothesis’, ‘Step’, and ‘Conclusion’ in the margin; WJEC examiners appreciate explicit signposting. If a proof involves a trigonometric identity, show the substitution explicitly. And if time permits, test your final formula with a small value of n to catch sign errors.

考试中为一个完整的归纳证明分配约 10–12 分钟。书写工整，每个关键等式单独成行。在页边空白处标注’基础情形’、’假设’、’步骤’、’结论’；WJEC 考官欣赏明确的路标指引。若证明涉及三角恒等式，要把代换过程清晰展示出来。如果时间允许，用一个小 n 值检验你最终的公式，以发现符号错误。

12. Practice Prompts for Self-Study | 自主练习题目

To consolidate your skills, attempt these WJEC-style proof questions: (1) Prove by induction that Σᵣ₌₁ⁿ r² = n(n+1)(2n+1)/6. (2) Show by induction that 3ⁿ − 1 is divisible by 2 for all n ∈ ℕ. (3) Using de Moivre’s theorem, express cos 3θ in terms of cos θ. For each, write a full essay-style solution and compare it with the structure outlined in this article. Regular practice will make the framework second nature.

为巩固技巧，请尝试以下 WJEC 风格的证明题：(1) 用归纳法证明 Σᵣ₌₁ⁿ r² = n(n+1)(2n+1)/6。(2) 用归纳法证明对所有 n ∈ ℕ，3ⁿ − 1 可被 2 整除。(3) 利用棣莫弗定理，用 cos θ 表示 cos 3θ。为每一题写出完整的论文式解答，并与本文概述的结构进行对照。经常练习将使这一框架成为你的第二天性。

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Writing a Mathematical Proof: Structure, Framework, and Model Essay for WJEC Further Mathematics | WJEC进阶数学论文写作：框架与范文