---
title: "Nth-Term Rules"
description: "Find the nth-term expression for an arithmetic sequence by identifying the common difference and the zero-term, and use it to determine any term in the sequence or test whether a given number belongs "
canonical: https://lightmysky.com/learn/mathematics/nth-term-rules-mt_HRKzwEQJgO
source: https://lightmysky.com/learn/mathematics/nth-term-rules-mt_HRKzwEQJgO.md
retrieved: 2026-10-19
---

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# Nth-Term Rules

Find the nth-term expression for an arithmetic sequence by identifying the common difference and the zero-term, and use it to determine any term in the sequence or test whether a given number belongs to the sequence

Subject: Mathematics · Area: Algebra · Ages 12 to 14
Page: https://lightmysky.com/learn/mathematics/nth-term-rules-mt_HRKzwEQJgO

## Ready when they can

- Derive the nth-term rule for an arithmetic sequence such as 3, 7, 11, 15, … as 4n − 1
- Use an nth-term formula to find the 50th or 100th term without listing all preceding terms
- Determine whether a given number (e.g., 99) is a term in a specified arithmetic sequence

## Lesson: Finding the nth-term rule

You already know how to build a sequence by adding the same number again and again. Now you'll turn that pattern into one formula, called the nth-term rule. Plug in a position number, n, and the formula hands you that term, no counting needed.

*(drawing: Every term here is 4 more than the one before it. That fixed jump is called the common difference.)*

**Example.** Take 3, 7, 11, 15. The common difference is 4, so the rule starts with 4n. Try n = 1: 4 x 1 = 4. The real first term is 3, one less than 4. So the rule is 4n - 1. Check n = 2: 4 x 2 - 1 = 7. It matches.

**Tip.** There's a shortcut for that extra number. Find the zero term: subtract the common difference from the first term. For 3, 7, 11, 15, that's 3 - 4 = -1, exactly the number sitting in 4n - 1.

**Example.** Once you have the rule, you can jump anywhere. For 4n - 1, the 50th term is 4 x 50 - 1 = 199, with no need to list 49 terms first. To check whether 99 belongs to this sequence, set 4n - 1 = 99. That gives 4n = 100, so n = 25. Since 25 is a whole number, 99 is the 25th term.

**Recap.** An nth-term rule is (common difference) x n + (zero term); use it to jump straight to any term or to test whether a number belongs to the sequence.

## 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)
- [Solving linear equations](https://lightmysky.com/learn/mathematics/solving-linear-equations-mt_QhFEDyIwSO)
- [Generating Sequences](https://lightmysky.com/learn/mathematics/generating-sequences-mt_yXO7lQ9Yn7)

## Opens up

- [Arithmetic Series and Sigma Notation](https://lightmysky.com/learn/mathematics/arithmetic-series-and-sigma-notation-mt_bAIPng87JO)
- [Quadratic Sequences and Second Differences](https://lightmysky.com/learn/mathematics/quadratic-sequences-and-second-differences-mt_gyZT3Tcw7U)
- [Geometric Sequences and Common Ratios](https://lightmysky.com/learn/mathematics/geometric-sequences-and-common-ratios-mt_NDnbJm2hTY)