---
title: "Generating Sequences"
description: "Generate terms of a sequence from a term-to-term rule (e.g., 'add 3 each time') or a position-to-term rule (e.g., '2n + 1'), and identify whether a sequence is arithmetic, geometric, or neither"
canonical: https://lightmysky.com/learn/mathematics/generating-sequences-mt_yXO7lQ9Yn7
source: https://lightmysky.com/learn/mathematics/generating-sequences-mt_yXO7lQ9Yn7.md
retrieved: 2026-10-19
---

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# Generating Sequences

Generate terms of a sequence from a term-to-term rule (e.g., 'add 3 each time') or a position-to-term rule (e.g., '2n + 1'), and identify whether a sequence is arithmetic, geometric, or neither

Subject: Mathematics · Area: Algebra · Ages 11 to 14
Page: https://lightmysky.com/learn/mathematics/generating-sequences-mt_yXO7lQ9Yn7

## Ready when they can

- Continue a sequence given a term-to-term rule involving addition, subtraction, or multiplication
- Generate the first five terms from a position-to-term formula such as 3n − 2
- Classify sequences as arithmetic (constant difference), geometric (constant ratio), or other

## Lesson: Generating sequences from a rule

A sequence is a list of numbers in order, and every sequence follows a rule. A term-to-term rule tells you how to get the next number from the one right before it, like 'add 3 each time'. A position-to-term rule tells you how to find any term directly from its position, using a formula like 2n + 1.

*(drawing: This sequence follows the term-to-term rule 'add 3 each time'.)*

**Example.** For a position-to-term rule, you substitute the position number, n, into the formula. Using 3n - 2: when n = 1, 2, 3, 4, 5, the terms are 1, 4, 7, 10, 13.

Sequences also come in types. In an arithmetic sequence, the difference between each term and the one before it stays the same, like always adding 3. In a geometric sequence, the ratio between each term and the one before it stays the same, like always multiplying by 2. A sequence that does neither is just called neither.

**Tip.** To find the type, subtract each term from the next one. If you always get the same number, it's arithmetic. If dividing each term by the one before it always gives the same number instead, it's geometric.

**Recap.** Build a sequence by repeating a term-to-term rule or substituting into a position-to-term formula, then check the differences or ratios to see if it's arithmetic, geometric, or neither.

## Practice

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

## Needs first

- [Linear number sequences](https://lightmysky.com/learn/mathematics/linear-number-sequences-mt_hBZwbst0ow)
- [Substituting into Formulae](https://lightmysky.com/learn/mathematics/substituting-into-formulae-mt_vHzVa3SURC)

## Opens up

- [Nth-Term Rules](https://lightmysky.com/learn/mathematics/nth-term-rules-mt_HRKzwEQJgO)