Leaving Cert Mathematical Induction: Proof by Induction Step by Step

Mathematical induction is a proof method with three steps: prove the base case, assume the statement for n = k, then prove it for n = k + 1. It is examined on Paper 1.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

Induction proves a statement is true for all natural numbers from a starting value.
The three standard types are series proofs, divisibility proofs and inequality proofs.
Every induction proof must clearly show the base case, the assumption and the inductive step.
Induction is examined on Paper 1.

Mathematical Induction explained

What is Mathematical Induction?

Mathematical induction is a powerful proof technique used to prove statements about natural numbers. It works like climbing a ladder:

Prove you can step onto the first rung (Base Case)
Prove that if you're on any rung, you can reach the next one (Inductive Step)
Conclude you can reach any rung (Conclusion)

This method allows us to prove infinite statements with just two steps!

The Three Steps of Induction

Every induction proof has three essential steps:

Step 1: Base Case Verify the statement is true for the smallest value (usually $n = 1$ ).

Step 2: Inductive Hypothesis Assume the statement is true for some value $n = k$ . This is our assumption.

Step 3: Inductive Step Using the assumption from Step 2, prove the statement is true for $n = k+1$ .

Conclusion: By the principle of mathematical induction, the statement is true for all natural numbers $n \geq 1$ .

When to Use Induction

Mathematical induction is ideal for proving statements that:

Involve natural numbers ( $n = 1, 2, 3, \ldots$ )
Have a pattern or formula
Claim something is true for all values

Common applications: - Summation formulas: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$ - Divisibility: " $7^n - 1$ is divisible by 6" - Inequalities: " $2^n > n$ for all $n \geq 1$ " - Sequence properties

If you see "for all $n \geq 1$ " or a formula with $n$ , think induction!

This is an introduction. The full Mathematical Induction lessons, worked solutions and practice questions are available to members. Start your free trial.

Key formulas

Name	Formula	Description
Induction Step 1	$P(1) \text{ is true}$	Prove the statement holds for n=1
Induction Step 2	$P(k) \Rightarrow P(k+1)$	Assume P(k) true, prove P(k+1) follows
Induction Conclusion	$\forall n \in \mathbb{N}, P(n)$	Statement holds for all natural numbers

Worked examples

Worked example 1

Which of the following is the correct ORDER of steps in a mathematical induction proof?

The correct order is: (1) Base Case - verify for the smallest value, (2) Inductive Hypothesis - assume true for n=k, (3) Inductive Step - prove true for n=k+1. This systematic approach ensures the proof is valid.

Answer:

Base Case, Inductive Hypothesis, Inductive Step

Worked example 2

What is the sum of the first 5 natural numbers using the formula?

Using the formula: sum = n(n+1)/2 = 5(6)/2 = 15. You can verify: 1+2+3+4+5 = 15.

Answer:

Where students lose marks

Skipping or rushing the base case.
Not clearly stating the inductive assumption for n = k.
Failing to show where the assumption is used in the n = k + 1 step.

Frequently asked questions

What are the steps of proof by induction?

First prove the base case (usually n = 1). Then assume the statement is true for n = k. Finally prove it follows for n = k + 1, using your assumption.

What types of induction questions come up?

The Leaving Cert examines series proofs, divisibility proofs and inequality proofs by induction.

Is mathematical induction on Paper 1?

Yes, mathematical induction is examined on Paper 1.

Authoritative sources

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