Why students lose marks in proof by induction and it is always the inductive step that is incomplete
Proof by mathematical induction is the only topic in IB AA HL where the structure of the answer is more important than the calculation inside it. A student who completes every algebraic step correctly but writes the result is true for all n without the closing statement of the inductive argument has not written a proof. They have written a sequence of correct equations. In 25 years of teaching, I have marked hundreds of induction proofs where the base case was correct, the inductive hypothesis was stated, the algebra was flawless, and the conclusion was absent or incomplete. The inductive argument closes with a specific sentence: Since the result is true for n equals k plus 1 whenever it is true for n equals k, and it is true for n equals 1, by the principle of mathematical induction it is true for all positive integers n. Every word in that sentence is structural. These worksheets make writing it automatic.
Recognition Training
Mathematical Induction
Proving summation formulas and divisibility statements by induction. The four steps are non-negotiable: base case, inductive hypothesis, inductive step, and conclusion. The most common error: jumping directly to the k plus 1 result without showing how it follows algebraically from the k assumption.
Direct Proof & Proof by Contradiction
Writing direct proofs using the algebraic definition of odd, even, and divisible integers. Proving irrationality by contradiction by assuming a rational form in lowest terms and deriving a contradiction. The most common error: writing a conclusion without algebraic justification, which earns no marks regardless of whether the result is correct.
Proof by Counterexample & Disproving Conjectures
Disproving universal statements by finding a single counterexample, and understanding why no finite number of examples can prove a universal statement. The most common error: testing several cases, finding them all true, and concluding the statement is proved. One counterexample disproves -- but a proof requires covering all cases.
The 4 Patterns Behind Every Lost Mark
Assuming the k plus 1 result in the inductive step
The inductive step must derive the result for n equals k plus 1 from the assumed result for n equals k. Students who begin by assuming what they are trying to prove have not written a valid inductive step.
Omitting the closing statement
The final sentence of every induction proof is structural and not optional. Ending the proof at the algebraic result without the conclusion costs the reasoning mark every time.
Incorrect base case
The base case must verify the statement for the smallest value in the domain. Students who verify n equals 1 when the domain begins at n equals 2 start in the wrong place. Always identify the first value in the domain before beginning.
Non-commutative error in matrix power proofs
Matrix multiplication is not commutative. The inductive step must maintain consistent ordering throughout. Students who swap the order of multiplication mid-calculation produce an invalid proof.
The Full Diagnostic Path
- 50 original exam-style questions across 5 sections
- Full worked solutions with M1/A1/R1 IB mark scheme
- Mistake analysis on every question
- Sections: summation formulas, divisibility proofs, inequalities, matrix powers, mixed proof types
- IB Examiner commentary per section
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