Proof by induction — AS & A Level Further Mathematics (9231) questions by topic
Recurrence relations and divisibility proofs anchor this topic: proving by induction that a sequence defined by u_1 = 5 and a linear recurrence equals 6^n - 1, and establishing that a sum of two large powers, offset by a constant, always divides exactly by a fixed integer for every positive whole number n.
Show and deduce are the two command words across 9 questions from 2023 to 2025 papers, worth from 5 to 7 marks, where the inductive step must explicitly use the assumed case rather than merely restating the result. With no examiner-report extracts logged for this topic, submitting each base-case-and-inductive-step write-up for instant marking is the way to confirm the induction is complete and logically closed before it is treated as proven.
May–June 2025, Paper 11
The sequence u1, u2, u3, ... is such that u1 = 5 and u{n+1} = 6un + 5 for n = 1. Prove by induction that un = 6^n - 1 for all positive integers n.
Answer this question and get it marked →May–June 2025, Paper 12
The sequence u1, u2, u3, ... is such that u1 = 5 and u{n+1} = 6un + 5 for n = 1. Prove by induction that un = 6^n - 1 for all positive integers n.
Answer this question and get it marked →May–June 2025, Paper 13
Prove by mathematical induction that 2025^n + 47^n - 2 is divisible by 46 for all positive integers n.
Answer this question and get it marked →May–June 2024, Paper 12
Prove by mathematical induction that 6^(4n) + 38^n - 2 is divisible by 74 for all positive integers n.
Answer this question and get it marked →October–November 2024, Paper 11
Prove by mathematical induction that, for all positive integers n, (d^n/dx^n)(tan^(-1) x) = Pn(x) (1 + x^2)^(-n), where Pn(x) is a polynomial of degree n - 1.
Answer this question and get it marked →October–November 2024, Paper 12
The sequence u1, u2, u3, ... is such that u1 = 4 and u(n+1) = 3un - 2 for n = 1. Prove by induction that un = 3^n + 1 for all positive integers n.
Answer this question and get it marked →October–November 2024, Paper 13
Prove by mathematical induction that, for all positive integers n, d^n/dx^n (tan^(-1) x) = Pn(x)(1 + x^2)^(-n), where Pn(x) is a polynomial of degree n - 1.
Answer this question and get it marked →May–June 2023, Paper 13
Prove by mathematical induction that, for all positive integers n, 5^(3n) + 32^n - 33 is divisible by 31.
Answer this question and get it marked →October–November 2023, Paper 12
Prove by mathematical induction that, for all positive integers n, (d^n/dx^n)(x^2 e^x) = (x^2 + 2nx + n(n-1)) e^x.
Answer this question and get it marked →Other AS & A Level Further Mathematics topics
- Matrices (38)
- Inference and hypothesis testing (31)
- Summation of series (28)
- Differential equations (24)
- Hooke's law and elasticity (21)
- Hyperbolic functions (18)
- Projectiles (18)
- Non-parametric tests (18)
- Complex numbers (17)
- Vectors (16)
- Equilibrium of a rigid body (16)
- Continuous random variables (16)