---
title: "From Examples to a General Argument"
description: "Move from checking a pattern on a few cases to an argument that covers every case, using a letter to stand for any number."
canonical: https://lightmysky.com/learn/mathematics/from-examples-to-a-general-argument-mt_vm7cigeRtg
source: https://lightmysky.com/learn/mathematics/from-examples-to-a-general-argument-mt_vm7cigeRtg.md
retrieved: 2026-09-12
---

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# From Examples to a General Argument

Move from checking a pattern on a few cases to an argument that covers every case, using a letter to stand for any number.

Subject: Mathematics · Area: Mathematical Thinking · Ages 12 to 13
Page: https://lightmysky.com/learn/mathematics/from-examples-to-a-general-argument-mt_vm7cigeRtg

## Ready when they can

- Say why testing five cases does not settle a claim about all numbers
- Turn a spotted pattern into a statement with a letter in it and check the statement against a fresh case
- Write a short argument for a claim such as the sum of two even numbers being even

## Lesson: From a few cases to every case

Checking five cases never settles a claim about all numbers. Five examples cover five numbers, while the claim covers infinitely many. One untested number could break the pattern, so examples can suggest but never prove.

**Example.** Turn a spotted pattern into a statement with a letter in it. Seeing 3 + 5 = 8 and 7 + 9 = 16 suggests a rule: the sum of two odd numbers is even. Or a figure pattern becomes 3n + 1 sticks for the nth figure. Then test it on a fresh case: n = 10 gives 31.

**Example.** A real argument covers every case at once. An even number is 2 times a whole number, so write two evens as 2a and 2b. Their sum is 2a + 2b, which equals 2(a + b). A multiple of 2 is even, for every a and b. Done for all of them.

**Tip.** Checking one fresh case supports a rule but never proves it. If Priya turns a pattern into 2n and n = 6 gives 12, the rule survives one test with the rest untested. Support is not proof. Proof leaves no case out.

**Recap.** Examples suggest, letters state, and only an argument covering every case proves.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Generalising with repeated reasoning](https://lightmysky.com/learn/mathematics/generalising-with-repeated-reasoning-mt_iyovKgZC1q)
- [Constructing mathematical arguments](https://lightmysky.com/learn/mathematics/constructing-mathematical-arguments-mt_j5YqQnN6xe)
- [Algebraic Notation](https://lightmysky.com/learn/mathematics/algebraic-notation-mt_KwdjWEmMNo)

## Opens up

- [Finding a Counterexample](https://lightmysky.com/learn/mathematics/finding-a-counterexample-mt_EoLyw6eP87)