plumiProof: AQA A‑Level Further Maths questions
Every AQA A-Level Further Maths exam question on A Proof, 2019–2025, with its mark scheme and examiner’s report.
Paper
All papers
Exam
All years
All 14 questions on Proof, as a list
- June 2025, Paper 1, Q10 (4 marks): Astrid is solving this mathematics problem: The series n S is defined by ℤ, ( ... = + + + + ∈ 2 4 6 2 n S n n ) ≥1 n Prove by induction that ( ) = +1 n S n n As
- June 2025, AS Paper 1, Q14 (8 marks): (a) The matrices A and B are given by A = [ 4 –3] –1 1 and B = [ 5 4 ] –3 –2 14 (a) (i) Find the matrices A–1 and B–1 [2 marks] 14 (a) (ii) Hence verify that (A
- June 2024, Paper 1, Q6 (4 marks): The sequence u1, u2, u3, ... is defined by _ u1 = 1 _ un+1 = un + 3n Prove by induction that for all integers n ≥ 1 _ un = 3 2 n2 – 3 2 n + 1 [4 marks]
- June 2024, AS Paper 1, Q12 (4 marks): Prove by induction that, for all n ∈ ℕ, the expression 5n – 2n is divisible by 3 [4 marks] ���������������������������������������������������������������������
- June 2023, Paper 2, Q12 (6 marks): The function f is defined by f (n) = 33n+1 + 23n+4 (n 2 Z+) Prove by induction that f (n) is divisible by 19 for n × 1 [6 marks]
- June 2023, AS Paper 1, Q13 (10 marks): (a) Prove by induction that, for all integers n ≥ 1, n r =1 r2 = 1 6 n(n + 1)(2n + 1) [4 marks] 13 (c) Use the formula in part (a) to write down a factorised
- June 2022, Paper 2, Q5 (4 marks): Prove by induction that, for all integers n × 1, X n r=1 r3 = 1 2 n(n + 1) 2 [4 marks]
- June 2022, AS Paper 1, Q11 (4 marks): Prove by induction that, for all integers n × 1 , (ABA−1)n = ABnA−1 where A and B are square matrices of equal dimensions, and A is non-singular. [4 marks]
- November 2021, Paper 1, Q5 (5 marks): The matrix M is defined by M = 3 2 −2 0 1 0 0 0 1 2 64 3 75 Prove by induction that Mn = 3n 3n − 1 −3n + 1 0 1 0 0 0 1 2 64 3 75 for all integers n × 1 [5 marks
- November 2021, AS Paper 1, Q13 (4 marks): Prove by induction that, for all integers n × 1 X n r=1 2−r = 1 − 2−n [4 marks]
- November 2020, Paper 2, Q10 (6 marks): The sequence u1, u2, u3, ... is defined by u1 = 0 un+1 = 5 6 × un Prove by induction that, for all integers n ≤ 1 , un = 5n × 5 5n × 1 [6 marks]
- November 2020, AS Paper 1, Q7 (4 marks): Prove by induction that, for all integers n × 1 , the expression 7n − 3n is divisible by 4 [4 marks]
- June 2019, Paper 2, Q10 (7 marks): Prove by induction that f (n) = n3 + 3n2 + 8n is divisible by 6 for all integers n × 1 [7 marks]
- June 2019, AS Paper 1, Q12 (12 marks): The matrix A is given by A = 1 2 0 3 12 (a) Prove by induction that, for all integers n ≤ 1, An = 1 3n − 1 0 3n [4 marks] 12 (b) Find all invariant lines under
Questions, mark schemes and examiner’s reports are © AQA, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every AQA A-Level Further Maths paper.
Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.