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AS Level · Further Pure Mathematics 1 WFM01

Proof by Induction

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Edexcel IAL · Further Pure 1

Proof by Induction

Show it works for one case, then show it ripples forward forever — and you've proved it for all positive integers.

4 Proof Types Covered

Worked Examples Included

Practice Questions

Exam Traps Flagged

Introduction

Series

Divisibility

Sequences

Matrices

Checklist & Tips

Chapter Overview

Proof by induction is a 4-step method to show a result is true for every positive integer (or integer ≥ some base).
Step 1 — Basic step: confirm the formula works for the starting value (usually n = 1 or n = 0).
Step 2 — Assumption step: assume the result is true for n = k (a general integer).
Step 3 — Inductive step: use that assumption to prove it's true for n = k + 1.
Step 4 — Conclusion: two precise sentences linking all three steps to clinch the proof.
The method works identically for four types of problems: series, divisibility, sequences, and matrices.
The conclusion wording is mark-scheme language — memorise it word-for-word.

1. What is Proof by Induction?

Proof by induction proves a statement is true for an infinite set of integers, starting from some base value, without checking each one individually. That sounds impossible — but the trick is brilliant in its simplicity.

🁢 The Domino Analogy

Imagine an infinite row of dominoes. You want all of them to fall. You don't need to push each one — you just need two things:
(1) Knock the first domino over.
(2) Show that whenever any domino falls, it knocks over the next one.
Those two facts guarantee every single domino falls. That's exactly how induction works.

The 4 Steps — Always in This Order

Basic Step

Verify the result is true for the base case, usually n = 1 or n = 0. Substitute directly into both sides and confirm they match.

Assumption Step

Write down clearly: "Assume the result is true for n = k", then state the formula with n replaced by k. No calculation here — just write it.

Inductive Step

This is the hard part. Start from the n = k+1 expression, manipulate it algebraically, and substitute in your Step 2 assumption to show the result holds for n = k+1.

Conclusion

Write exactly these two sentences: "If it is true for n = k, then it is true for n = k + 1. As it is true for n = 1, the statement is true for all n ∈ ℤ⁺."

Critical: The conclusion step earns its own mark. Lose the exact wording, lose the mark. Memorise it.

The Conclusion — Memorise Word for Word

"If it is true for n = k, then it is true for n = k + 1."
"As it is true for n = 1, the statement is true for all n ∈ ℤ⁺."
(Replace n = 1 with whatever base case you used.)

Proof Type 1

2. Proof by Induction — Series (Summation)

You are given a closed-form formula for a sum and asked to prove it equals that formula for all positive integers. The key technique in Step 3 is pulling out the last term from the sum.

Key Technique for Series — Step 3

factorise before

Worked Example — Series

Prove by induction:

Proof Type 2

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Proof by Induction

Proof by Induction

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