Topic summary · Further maths (FP1)

Proof by Induction summary

Induction proves a statement for all integers $n\ge n_0$ : base case, inductive assumption, inductive step, conclusion.

Open Learning Sheet — Proof by Induction Download Proof by Induction answers PDF

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Key learning points

Verify the base case carefully.
Assume true for $n=k$ ; prove true for $n=k+1$ .
Write an explicit conclusion invoking induction.
Common targets: sums, divisibility, inequalities, matrix powers.

Worked example 1 — Odd + odd

Prove the sum of two odds is even.

Working

(2k+1)+(2m+1)=2(k+m+1)

Algebraic parity.

Worked example 2 — Consecutive product

Show $n(n+1)$ is even.

Working

\text{one of }n,n+1\text{ is even}

Consecutive integers.

Worked example 3 — Square difference

Expand $(n+1)^2-n^2$ .

Working

2n+1

Next odd number.

Worked example 4 — Structure

Name the four ingredients of induction.

Working

\text{base; assume }k\text{; prove }k+1\text{; conclude}

Examiners look for all four spelled out.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.

Open Learning Sheet Answers PDF

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