Linear nth terms

Objective. Derive the nth term of an arithmetic sequence.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

A linear nth term is a formula that gives any term of an arithmetic sequence straight from its position number, without listing earlier terms. For a sequence with common difference d and first term a, the rule is an = dn + (a − d); find d by subtracting consecutive terms, then check against n = 1. For 5, 8, 11, 14, … the difference is 3, so the rule starts 3n, and testing n = 1 gives 3 + 2 = 5, so the rule is 3n + 2. Always verify your formula against at least two terms before trusting it.

Worked example

Find the nth term of the linear sequence 7, 11, 15, 19, …

1 $Find the common difference: 11 - 7 = 4$
2 $Start with 4n as the base term$
3 $Substitute n = 1: 4(1) = 4, but the first term is 7, so add 3$
4 $The nth term is 4n + 3$

Answer: 4n + 3

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Pupils often use the first term as the constant instead of adjusting it, writing 3n + 5 for 5, 8, 11, 14 because 5 is the first term. Always substitute n = 1 into your formula and check it equals the actual first term.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard