From one statement to infinitely many A statement indexed by an integer is a family of claims. Writing \(P(n)\) gives one name to the entire family. Mathematical induction proves every claim from a starting index without checking them one by one. The domino bridge The base case stands up the first domino. The inductive step proves that whenever an arbitrary domino \(k\) falls, the next domino \(k+1\) must fall. Both facts are needed. \[P(n_0)\text{ is true}\qquad\text{and}\qquad \forall k\ge n_0,\ P(k)\Rightarrow P(k+1)\] StageWhat to writeLogical purpose DefineLet \(P(n)\) be the stated claim for \(n\ge n_0\).Fixes the exact proposition and domain. Base caseSubstitute the smallest admissible value \(n_0\).Starts the chain. HypothesisAssume \(P(k)\) is true for some integer \(k\ge n_0\).Provides one temporary, arbitrary link. Inductive stepUsing the hypothesis, prove the exact statement \(P(k+1)\).Shows every link forces the next. ConclusionState that \(P(n)\) is true for all integers \(n\ge n_0\).Connects the two established facts to the required domain. Exam check The induction hypothesis is not the conclusion. It is assumed only for one arbitrary index \(k\) in order to prove the next index. Never write “assume true for all \(k\)”. Non-negotiable details Use the smallest value stated in the question; it need not be \(0\) or \(1\). In the base case, compare the actual left-hand and right-hand sides. State “for some \(k\)” in the hypothesis and “for all \(n\)” only in the final conclusion. Write the target form of \(P(k+1)\) before beginning the algebra. Class check For the claim \(3^n>n^2\) for \(n\ge3\), write \(P(3)\), the induction hypothesis \(P(k)\) and the exact target \(P(k+1)\) without attempting the proof.

The Logical Contract of Induction

IB Year 5 | Grade 11

From one statement to infinitely many

A statement indexed by an integer is a family of claims. Writing \(P(n)\) gives one name to the entire family. Mathematical induction proves every claim from a starting index without checking them one by one.

The domino bridge

The base case stands up the first domino. The inductive step proves that whenever an arbitrary domino \(k\) falls, the next domino \(k+1\) must fall. Both facts are needed.

\[P(n_0)\text{ is true}\qquad\text{and}\qquad \forall k\ge n_0,\ P(k)\Rightarrow P(k+1)\]

Stage	What to write	Logical purpose
Define	Let \(P(n)\) be the stated claim for \(n\ge n_0\).	Fixes the exact proposition and domain.
Base case	Substitute the smallest admissible value \(n_0\).	Starts the chain.
Hypothesis	Assume \(P(k)\) is true for some integer \(k\ge n_0\).	Provides one temporary, arbitrary link.
Inductive step	Using the hypothesis, prove the exact statement \(P(k+1)\).	Shows every link forces the next.
Conclusion	State that \(P(n)\) is true for all integers \(n\ge n_0\).	Connects the two established facts to the required domain.

Exam check

The induction hypothesis is not the conclusion. It is assumed only for one arbitrary index \(k\) in order to prove the next index. Never write “assume true for all \(k\)”.

Non-negotiable details

Use the smallest value stated in the question; it need not be \(0\) or \(1\).
In the base case, compare the actual left-hand and right-hand sides.
State “for some \(k\)” in the hypothesis and “for all \(n\)” only in the final conclusion.
Write the target form of \(P(k+1)\) before beginning the algebra.

Class check

For the claim \(3^n>n^2\) for \(n\ge3\), write \(P(3)\), the induction hypothesis \(P(k)\) and the exact target \(P(k+1)\) without attempting the proof.

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