Proof by Mathematical Induction
Proving statements about series, divisibility and matrix powers through a base case, a clearly stated hypothesis and a valid inductive step.
How to study A-level Further Mathematics
Define the objects and conditions, select a representation, carry out exact mathematics, then validate, interpret and communicate the result.
Core concepts
Concept 1
Induction proves a statement for every permitted integer by linking a verified base case to the next case.
Exam cue: Write the proposition as P(n) and verify the smallest permitted value.
Concept 2
The inductive hypothesis must be used explicitly rather than merely restated.
Exam cue: Assume P(k) for an arbitrary permitted k, then derive P(k + 1).
Concept 3
The base value and step must match the claimed domain, including statements beginning above one.
Exam cue: Close by stating exactly which integer values follow by induction.
Risk pitfalls and guardrails
Assuming the result for k + 1.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Omitting the base case.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Algebraically checking two consecutive examples instead of proving the step.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Memory anchors
Base Case
The base case verifies the proposition at the first value in its claimed domain.
Inductive Hypothesis
The inductive hypothesis assumes P(k) for an arbitrary permitted integer k.
Inductive Step
The inductive step uses P(k) to establish P(k + 1).
Arbitrary k
The integer k is arbitrary, so the step is not tied to one numerical example.
Induction Conclusion
The conclusion links the base case and inductive step to every claimed integer.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
To prove 1+3+⋯+(2n−1)=n² for n≥1 by induction, what expression begins the k→k+1 step?
In an induction proof for 2ⁿ≥n+1, n≥0, which base case is required?
Answer all questions to submit.
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Move forward only after this module is stable.
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