Leaving Cert Maths: Proof by induction
How often Proof by induction comes up on the Maths papers, every year it was asked, and questions to try.
HL Asked on 6 of the last 10 Higher Level papers, most recently in 2026. most years
Proof by induction, Higher Level(6 marks)
Quick ones on Proof by induction.
(a)The first step in a proof by induction is to …
- assume it true for n = k
- prove it true for n = 1
- prove it true for n = k + 1
(b)In the inductive step, you assume the statement is true for …
- all n
- n = k + 1
- n = k
(c)Proof by induction proves a statement for all …
- natural numbers n
- complex numbers z
- real numbers x
Show the answers
(a) prove it true for n = 1
(b) n = k
(c) natural numbers n
Higher Level
Asked on 6 of the last 10 Higher Level papers, most recently in 2026. most years
Every paper, year by year
| Year | Where it came up |
|---|---|
| 2026 | Q6 |
| 2025 | Q10 |
| 2024 | Not asked |
| 2023 | Not asked |
| 2022 | Not asked |
| 2021 | Q4 |
| 2019 | Q2 |
| 2018 | Q4 |
| 2017 | Not asked |
| 2016 | Q4 |
Links open the State Examinations Commission’s paper for that year.
More Proof by induction questions
Proof by induction, 2 marks
1 + 3 + 5 + … + (2n − 1) = ?
- 2n²
- n(n + 1)
- n²
Show the answer
n²
It's the sum of the first n odd numbers. Arithmetic series: (n/2)(1 + (2n − 1)) = (n/2)(2n) = n².
Proof by induction, 2 marks
For all n ∈ ℕ, 4ⁿ − 1 is divisible by …
- 5
- 3
- 4
Show the answer
3
For n = 1: 4 − 1 = 3. Then 4ᵏ⁺¹ − 1 = 4(4ᵏ − 1) + 3, which is divisible by 3 if 4ᵏ − 1 is.