Proof (ME-P1)

4 dot points across 4 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we use induction to prove general statements involving a recursion, a formula or a property?

Apply mathematical induction to prove general statements about a recursive sequence, a property of a formula, or a recursive procedure

Induction for general statements beyond sums, divisibility and inequalities. Closed-form proofs for recursively defined sequences, properties preserved by an iterative process, a note on strong induction and choosing the induction variable, and worked linear and non-linear recurrences plus a counting result.

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How do we use mathematical induction to prove identities involving sums of a sequence?

Prove identities for sums of series using the principle of mathematical induction

Induction proofs of series identities. The principle as a chain of dominoes, the standard four-part structure, the split-off-the-last-term technique that drives every inductive step, the four sum formulas worth memorising, and fully worked proofs for sums of integers, squares, cubes, geometric series and partial-fraction telescoping sums.

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How do we use induction to prove an inequality holds for every positive integer?

Prove inequalities involving an integer parameter

n

using mathematical induction

Induction for inequalities. The four-part structure, the use-then-strengthen two-step that turns the hypothesis into the n = k + 1 inequality, the growth-comparison view, choosing the right base case, and fully worked proofs including 2^n greater than n squared, n factorial beating 2^n, and a sum inequality.

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How do we use mathematical induction to prove divisibility statements?

Prove divisibility statements involving an integer parameter

n

using mathematical induction

Induction proofs of divisibility. The four-part structure, the add-and-subtract trick that rewrites the n = k + 1 expression in terms of the n = k case plus a divisible chunk, the multiplier method for exponentials, and fully worked proofs including a mixed-base example and divisibility by a composite number.

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