How to Write A-Level Further Mathematics Proofs: The Four Steps of Mathematical Induction Are Essential, and How to Answer Proof by Contradiction and Show That Questions

"Proof questions" are the most frustrating type of problem for many students in China studying Further Mathematics: they can calculate, but they don't know how to "write" the proof. Math classes in China focus more on getting the final answer, whereas A-Level Further Mathematics proof questions test your ability to clearly and completely write out every step of your reasoning. Based on the Cambridge 9231 Further Mathematics (2026 and 2027 editions) and Edexcel IAL Mathematics syllabus (3rd edition), this article summarizes the proof methods required by both exam boards, with a focus on how to write mathematical induction proofs for full marks. For the overall structure of Further Mathematics, see How A-Level Further Mathematics is Assessed.

1. What Proofs Are Tested by Both Exam Boards

It is clear that mathematical induction is a core proof method shared by both exam boards for Further Mathematics; Edexcel also systematically teaches proof by exhaustion, counterexamples, and proof by contradiction in regular Mathematics. The phrase "not listed separately" in the Cambridge column above means we did not find corresponding entries in the content lists of the two syllabuses.

2. How the Exam Boards Officially Define "Prove" and "Show that"

Command words from the Cambridge 9231 syllabus:

• Prove: Confirm that a given statement is true using a sequence of logical mathematical reasoning;
• Show (that): Present organized evidence to arrive at a given result;
• Derive: Obtain another expression, equation, or value from a given one through a series of logical steps;
• Verify: Confirm that a given statement or result is true;
• Deduce: Draw a conclusion from given information.

These questions share a common feature: the answer is already given in the question. Therefore, examiners are not looking at the final result, but rather whether the reasoning process is complete and whether every step is justified. The Cambridge syllabus also requires candidates to write out all necessary working; answers consisting only of calculator output without supporting working will receive no marks.

Edexcel P2 describes the structure of a proof as: starting from given assumptions, proceeding through a series of logical steps, and reaching a conclusion.

3. Mathematical Induction: Syllabus Requirements and Question Types

Cambridge 9231 Section 1.7 requires:

• Prove given statements using mathematical induction. Examples in the syllabus cover sum formulas (e.g., sum of r³), general terms of sequences (sequences given by recurrence relations), nth powers of matrices, and divisibility;
• Recognize that "making a conjecture through a few trial calculations and then proving it by induction" is a useful strategy, and apply it in simple cases. Syllabus examples include finding the nth derivative of x eˣ and the sum of the first n terms of r × r!.

Edexcel FP1 requires induction proofs covering four categories: series summation, divisibility, general terms of sequences, and matrix products, which is largely consistent with Cambridge.

4. How to Write a Complete Induction Proof

Below is an example demonstrating the structure (this example is written by us and is not an exam question):

Proposition: For all positive integers n, 7ⁿ − 1 is divisible by 6.

1. Base case: When n = 1, 7¹ − 1 = 6, which is divisible by 6, so the proposition holds.
2. Inductive hypothesis: Assume the proposition holds for n = k (where k is a positive integer), i.e., 7ᵏ − 1 = 6m, where m is an integer.
3. Inductive step: When n = k + 1,
   7ᵏ⁺¹ − 1 = 7 × 7ᵏ − 1 = 7(7ᵏ − 1) + 7 − 1 = 7 × 6m + 6 = 6(7m + 1).
   Since 7m + 1 is an integer, 7ᵏ⁺¹ − 1 is divisible by 6, meaning the proposition also holds for n = k + 1.
4. Conclusion: The proposition holds for n = 1; and if it holds for n = k, it also holds for n = k + 1. Therefore, by mathematical induction, the proposition holds for all positive integers n.

Key Writing Tips (Based on Experience):

• Do not skip any of the four parts, especially the final concluding sentence; many students stop writing right after the inductive step;
• Write the inductive hypothesis as an "assumption", not as "since the proposition holds for k";
• Explicitly use the hypothesis in the inductive step, and indicate in the equations which step uses the hypothesis;
• Write out the target expression first: In summation questions, write out the expression to be proved for n = k + 1 first, then derive from the left side to the right side for a clearer direction;
• Matrix questions: Pay attention to the order of multiplication, Aᵏ⁺¹ = Aᵏ × A or A × Aᵏ; choose one and write it clearly;
• General term of sequence questions: Combine the recurrence relation with the hypothesis and simplify to the target form.

5. Other Proof Methods (Edexcel P2, P4)

• Proof by exhaustion: Verify all cases one by one. A syllabus example is: x and y are odd numbers less than 7, prove their sum is divisible by 2;
• Counterexample: Only one counterexample is needed to disprove a proposition. A syllabus example is: show that the proposition "n² − n + 1 is prime for all n" is false;
• Proof by contradiction: Assume the conclusion is false and derive a contradiction. Classic examples specified in the syllabus include proving √2 is irrational and that there are infinitely many primes; it also requires applying this method to unfamiliar proofs.

Key Writing Tips (Based on Experience): For proof by contradiction, the first sentence must clearly state "Assume...", and the conclusion must point out where the contradiction lies and draw a conclusion; when providing a counterexample, substitute the counterexample, calculate the specific result, and explain why it does not satisfy the proposition.

6. "Show that" Questions: Do Not Skip Steps

A large number of questions in Further Mathematics use "Show that", such as proving hyperbolic function identities, deriving trigonometric identities, or simplifying to a given result. Here are some tips based on experience:

• Work from one side to the other: When proving an identity, start from the more complicated side and simplify step by step to the other side; do not manipulate both sides simultaneously and claim they "look equal";
• Write out key steps: Since the result is known, skipped steps can easily be seen as "fudged";
• Match the last line with the question: The final line of your derivation must exactly match the result given in the question;
• Use "hence": When the question says "Hence...", you must use the result from the previous part, rather than starting from scratch.

7. Common Reasons for Losing Marks

Here is a summary based on experience:

• Missing concluding sentence in induction, or only writing "obviously holds" for the base case;
• Not using the hypothesis in the inductive step, or treating the conclusion to be proved as a given;
• Skipping algebraic steps: Writing the final form directly in the middle;
• Failing to clearly state the assumption and contradiction in proof by contradiction;
• Manipulating both sides simultaneously in "Show that" questions.

8. How to Practice

1. Write out the syllabus examples one by one: The examples in Cambridge syllabus Section 1.7 and Edexcel FP1 cover four categories: summation, sequences, matrices, and divisibility. Write out a complete proof for each category;
2. Practice "conjecture then prove": Required by the Cambridge syllabus, for example, finding the nth derivative of x eˣ: calculate the first few derivatives, find the pattern, and then prove it by induction;
3. Refer to mark schemes: Mark schemes for past papers are usually obtained through your school. Check how many marks are allocated for each part of an induction proof;
4. Students applying for math-related majors: TMUA Paper 2 tests logic and proof, and STEP tests long-form proofs. See Math Admissions Tests TMUA and STEP.

Conclusion

Proof questions in Further Mathematics test your complete and rigorous written reasoning: all four parts of mathematical induction are essential, proof by contradiction requires clearly stating the assumption and contradiction, and "Show that" questions must not skip steps. Both Cambridge and Edexcel list mathematical induction as a core component of Further Mathematics. By practicing the syllabus examples thoroughly for each category, you can turn proof questions from "knowing how to do it but losing marks" into a steady source of points. If you need A-Level Further Mathematics tutoring, feel free to chat with our teachers.

Verification Note: This article was verified in September 2026: The requirements and examples for mathematical induction in Cambridge 9231 Section 1.7, the proof requirements for hyperbolic function identities and eigenvalue properties, the proof of de Moivre's theorem for positive integer indices, the meanings of the command words Prove, Show that, Derive, Verify, and Deduce, and the rule that "answers with only calculator output and no working will receive no marks" are from Cambridge International's "Further Mathematics 9231 syllabus for 2026 and 2027"; the fact that proof methods are not listed separately in the Cambridge 9709 content list is from "Mathematics 9709 syllabus for 2026 and 2027". The proof structure, proof by exhaustion and counterexamples in Edexcel P2, proof by contradiction in P4, the four categories of mathematical induction questions in FP1, and the proof of de Moivre's theorem for any integer in FP2 are from the Pearson IAL Mathematics, Further Mathematics, and Pure Mathematics syllabuses (3rd edition). The induction proof example in this article is written by us and is not an exam question. The writing tips, common reasons for losing marks, and practice methods are our experience-based suggestions.

References:

• Cambridge International：AS & A Level Further Mathematics (9231)
• Pearson：International Advanced Level Mathematics