Proof

OCR

GCSE

Proof in Further Mathematics necessitates the construction of rigorous logical arguments to establish the universal validity of mathematical conjectures. Candidates must demonstrate mastery of formal methods, specifically Proof by Mathematical Induction, Proof by Contradiction, and Disproof by Counter-example, applied across diverse contexts such as number theory, matrices, and series. Success requires precise algebraic manipulation, the correct use of logical connectives, and the ability to structure arguments where every step is explicitly justified by established axioms or prior results.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Subtopics in this area

Proof

What You Need to Demonstrate

Key skills and knowledge for this topic

Award B1 for correctly verifying the base case (usually n=1) with explicit substitution
Award M1 for stating the inductive hypothesis: 'Assume true for n=k'
Award M1 for attempting the inductive step n=k+1 using the assumption
Award A1 for correct algebraic manipulation leading to the required form for k+1
Award A1 for the final completion statement: 'True for n=1, and if true for n=k then true for n=k+1, therefore true for all positive integers n'
Award M1 for correctly defining algebraic terms (e.g., 'Let 2n be an even integer' or 'Let n and n+1 be consecutive integers')
Award M1 for correct algebraic manipulation, such as expanding double brackets or factorising expressions fully
Award A1 for reaching a convincing intermediate form (e.g., showing '4(n^2 + n)' to demonstrate divisibility by 4)

Example Examiner Feedback

Real feedback patterns examiners use when marking

"You have correctly shown the algebra for n=k+1, but you missed the final concluding statement required for the full marks."
"Check your matrix multiplication order; remember that matrix multiplication is not commutative."
"You assumed the result is true for n=k, but you didn't explicitly state this assumption in writing."
"Excellent algebraic structure, but ensure you define any new integers introduced (e.g., 'where M is an integer') for divisibility proofs."
"You have verified this for n=1 and n=2, but a proof must work for *all* values — use 'n' instead of numbers"
"Your algebra is correct, but you lost the final mark. Add a sentence starting with 'Therefore...' to explain what your result shows"
"Do not move terms from LHS to RHS. Start with the LHS and manipulate it until it looks exactly like the RHS"
"To prove this is always positive, try completing the square rather than just substituting values"

Marking Points

Key points examiners look for in your answers

Award B1 for correctly verifying the base case (usually n=1) with explicit substitution
Award M1 for stating the inductive hypothesis: 'Assume true for n=k'
Award M1 for attempting the inductive step n=k+1 using the assumption
Award A1 for correct algebraic manipulation leading to the required form for k+1
Award A1 for the final completion statement: 'True for n=1, and if true for n=k then true for n=k+1, therefore true for all positive integers n'
Award M1 for correctly defining algebraic terms (e.g., 'Let 2n be an even integer' or 'Let n and n+1 be consecutive integers')
Award M1 for correct algebraic manipulation, such as expanding double brackets or factorising expressions fully
Award A1 for reaching a convincing intermediate form (e.g., showing '4(n^2 + n)' to demonstrate divisibility by 4)
Award A1/B1 (dependent) for a rigorous concluding statement that explicitly links the final algebraic form to the required proof (e.g., 'Since k is an integer, 2k+1 is always odd')

Examiner Tips

Expert advice for maximising your marks

💡Memorise the OCR-approved conclusion template for induction; losing the final A1 mark due to wording is a preventable error
💡When proving divisibility, explicitly write f(k+1) - f(k) or f(k+1) - m*f(k) to isolate the divisible term
💡For 'Show that' questions involving proof, you must show every line of working; skipping algebraic steps will result in loss of marks
💡In proof by contradiction, clearly state the negation of the statement at the very start
💡When asked to prove an expression is 'positive for all real values', immediately think of completing the square to show the result is of the form (x+a)^2 + b where b > 0
💡Never skip steps in algebraic expansion; examiners need to see the unsimplified expansion to award Method marks if the final answer is incorrect
💡Memorise the standard definitions: Even = 2n, Odd = 2n+1, Consecutive = n, n+1. Using 'n' for an odd number is a fatal error

Common Mistakes

Pitfalls to avoid in your exam answers

Omitting the 'if... then...' logic in the final conclusion, simply stating 'true for all n' without justifying the inductive link
In matrix induction, incorrectly assuming commutativity (e.g., writing M^(k+1) = M^k * M and M * M^k interchangeably without justification)
For divisibility proofs, failing to define an integer multiplier (e.g., writing f(k) = 7m without stating 'where m is an integer')
Algebraic errors when factorising the (k+1) expression, particularly with indices or factorials
Substituting specific numerical values (e.g., n=1, n=2) to 'prove' a general statement, which earns zero credit for the proof itself
Working across the equals sign when proving an identity (treating it like an equation to solve) rather than manipulating one side to match the other
Omitting the final conclusion; candidates often perform the algebra correctly but fail to state 'Therefore, the expression is a multiple of 9'
Circular reasoning: assuming the statement is true at the start of the proof to derive a truth (e.g., 0=0)

Study Guide Available

Comprehensive revision notes & examples

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Key Terminology

Essential terms to know

Proof by Mathematical Induction (Series, Divisibility, Matrices)

Proof by Contradiction (Irrationality, Infinity of Primes)

Disproof by Counter-example

Deduction and Exhaustion

Mathematical Induction (Series, Divisibility, Matrices)

Proof by Deduction

Proof by Contradiction

Disproof by Counter-example

Logical Connectives and Set Notation

Likely Command Words

How questions on this topic are typically asked

Prove

Show that

Deduce

Determine

Verify

Explain

Find

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