---
title: "Generalising with repeated reasoning"
description: "Recognise and use repeated reasoning to generalise: describe algebraic rules for nth terms, use properties of operations to simplify, and verify generalisations with specific cases"
canonical: https://lightmysky.com/learn/mathematics/generalising-with-repeated-reasoning-mt_iyovKgZC1q
source: https://lightmysky.com/learn/mathematics/generalising-with-repeated-reasoning-mt_iyovKgZC1q.md
retrieved: 2026-10-19
---

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# Generalising with repeated reasoning

Recognise and use repeated reasoning to generalise: describe algebraic rules for nth terms, use properties of operations to simplify, and verify generalisations with specific cases

Subject: Mathematics · Area: Mathematical Thinking · Ages 10 to 11
Page: https://lightmysky.com/learn/mathematics/generalising-with-repeated-reasoning-mt_iyovKgZC1q

## Ready when they can

- Explain that the interior angle sum of an n-sided polygon is (n−2) × 180° based on the pattern for triangles, quadrilaterals, pentagons
- Predict the 20th term of a linear sequence by identifying and applying the general rule
- Generalise that dividing by a non-zero whole number n always gives a denominator of n in the fraction

## Lesson: Finding the rule that always works

A pattern that works in a few examples often works for every case. Finding a rule like that is called generalising. You use a letter, like n, to stand for any number, so your rule works even for cases you have not tried.

*(drawing: Every extra side adds another 180° to the angle sum. What is the rule for n sides?)*

**Example.** Maya's sequence is 5, 9, 13, 17, and it goes up by 4 each time. Instead of counting all the way to the 20th term, she writes a rule: multiply the position number by 4, then add 1. For the 20th term: 4 × 20 + 1 = 81. She checks the rule on a term she already knows, term 3: 4 × 3 + 1 = 13. It matches, so the rule works.

*(drawing: Test the rule on a known term. 4x3+1=13 matches term 3. If it fails, fix the rule.)*

Dividing a whole number by a non-zero whole number splits it into equal parts. The number you divide by goes on the bottom, and you cannot divide by 0. So 3 divided by 5 is 3/5, and 9 divided by 4 is 9/4. Dividing a unit fraction by a whole number multiplies the bottom. So 1/4 divided by 2 is 1/8.

To go from an angle sum back to sides, undo the rule. For 1080: divide by 180 to get 6, then add 2 to get 8 sides.

**Recap.** Repeated reasoning turns a pattern from a few examples into a rule for any case, then you check the rule against numbers you already know.

## Practice

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

## Needs first

- [Angles in triangles (age 10+)](https://lightmysky.com/learn/mathematics/angles-in-triangles-age-10-mt_cdMlC7EpTJ)
- [Describing Rules & Patterns](https://lightmysky.com/learn/learning-to-learn/describing-rules-and-patterns-mt_hbe_kdE_7C)
- [Linear number sequences](https://lightmysky.com/learn/mathematics/linear-number-sequences-mt_hBZwbst0ow)
- [Reasoning with Equivalences](https://lightmysky.com/learn/mathematics/reasoning-with-equivalences-mt_jCy07DyBNU)

## Opens up

- [Nth-Term Rules](https://lightmysky.com/learn/mathematics/nth-term-rules-mt_HRKzwEQJgO)
- [From Examples to a General Argument](https://lightmysky.com/learn/mathematics/from-examples-to-a-general-argument-mt_vm7cigeRtg)