Multi-Step Algebraic Proof

At A-Level, many proof questions require you to construct arguments that span multiple steps, combining several algebraic techniques within a single proof. These questions test your ability to chain logical arguments together, apply identities and manipulations strategically, and present a coherent mathematical argument from start to finish.

Multi-step algebraic proofs frequently appear in AQA exam papers at the higher end of the mark range (5–8 marks). They assess AO2 (use and apply standard techniques) and AO1 (use and apply standard procedures) assessment objectives.

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Chaining Arguments

A multi-step proof involves linking several smaller results or manipulations together. Each step must follow logically from the previous one. The key is to plan your approach before you begin writing.

Example 1: Prove that for all integers n, (2n + 3)² − (2n − 1)² is a multiple of 8.

Step 1: Expand each square.

(2n + 3)² = 4n² + 12n + 9
(2n − 1)² = 4n² − 4n + 1

Step 2: Subtract.

(2n + 3)² − (2n − 1)² = (4n² + 12n + 9) − (4n² − 4n + 1)
                       = 16n + 8