The nth term: how to find it for a linear sequence

What is the nth term?

The nth term is a formula for a sequence that uses the letter n to stand for the position of a term. Put in n = 1 and you get the 1st term, put in n = 2 and you get the 2nd term, and so on. It's also called the nth term formula or the position-to-term rule.

Take the nth term 3n + 2. The 3n means 3 × n.

Position (n)	1	2	3	4	10
3n	3	6	9	12	30
3n + 2	5	8	11	14	32

The middle row is the 3 times table. The sequence 5, 8, 11, 14 is just the 3 times table with 2 added to every number. That's the idea behind every linear nth term: a times table, moved up or down.

Term-to-term and position-to-term rules

A term-to-term rule tells you how to get from one term to the next. For 5, 8, 11, 14 it's "add 3". Children describe sequences like this at primary school: the national curriculum has Year 5 children finding term-to-term rules and Year 6 maths includes generating and describing linear number sequences.

A position-to-term rule, the nth term, goes straight from the position to the term. To find the 100th term with the term-to-term rule, you'd add 3 ninety-nine times. With the nth term, the 100th term of 3n + 2 is 3 × 100 add 2, which is 302.

How to find the nth term of a linear sequence

To find the nth term of a linear sequence, find the difference, then work out what to add or take away. Linear sequences are also called arithmetic sequences or arithmetic progressions.

1. Find the difference between the terms. This is the number in front of n.
2. Write out that times table under the sequence.
3. Compare. Work out what you add to, or take away from, each number in the times table to get the term. That's the number on the end.

Worked example: 7, 11, 15, 19, 23

The terms go up by 4 each time, so the nth term starts with 4n.

n	1	2	3	4	5
4n	4	8	12	16	20
Sequence	7	11	15	19	23

Each term is 3 more than the 4 times table: 4 + 3 = 7 and 8 + 3 = 11. So the nth term is 4n + 3.

Check it with a term you didn't use: for n = 5, 4 × 5 = 20, and 20 + 3 = 23. That matches.

Worked example: a sequence that goes down

Find the nth term of 20, 17, 14, 11, 8.

The terms go down by 3 each time, so the difference is −3 and the nth term starts with −3n. The −3 times table goes −3, −6, −9, −12, −15. To get from −3 to 20 you add 23, and the same works for every term: −6 + 23 = 17. So the nth term is −3n + 23, which is usually written 23 − 3n.

Check: for n = 4, 3 × 4 = 12, and 23 − 12 = 11. That's the 4th term.

Worked example: is a number in the sequence?

Is 100 a term of the sequence with nth term 3n + 2?

If it is, then 3n + 2 = 100. Take 2 from both sides: 3n = 98. But 98 isn't in the 3 times table, so n isn't a whole number. There's no 32.67th term, so 100 is not in the sequence. The 32nd term is 98 and the 33rd is 101. This uses the same steps as solving equations.

Finding the sequence from the nth term

To find the sequence from the nth term, substitute n = 1, 2, 3 and so on. For 5n − 4:

• n = 1: 5 × 1 = 5, and 5 − 4 = 1
• n = 2: 5 × 2 = 10, and 10 − 4 = 6
• n = 3: 5 × 3 = 15, and 15 − 4 = 11

So the sequence is 1, 6, 11, 16, and so on. Substituting numbers into expressions is part of algebra basics.

How to find the nth term of a geometric sequence

A geometric sequence multiplies by the same number each time instead of adding: 3, 6, 12, 24 doubles every time. The key stage 3 programme of study asks children to recognise geometric sequences, and GCSE includes simple geometric progressions. But the curriculum only asks for nth term expressions for linear sequences and, at GCSE, quadratic sequences, so the method on this page is for linear sequences only.

Quadratic sequences, where the differences between the terms change by the same amount each time (2, 5, 10, 17, 26), are GCSE content. The key stage 4 programme of study lists finding their nth term as extra content for more highly attaining pupils.

Common mistakes with the nth term

• Writing the term-to-term rule as the nth term. For 5, 8, 11, 14, the answer is not n + 3. "Add 3" becomes 3n.
• Using the first term as the number on the end. For 5, 8, 11, 14, it isn't 3n + 5. Compare with the times table: 3 + 2 = 5, so it's 3n + 2.
• Losing the minus sign. If the sequence goes down, the number in front of n is negative.
• Mixing up position and term. In 3n + 2, n is the position (1st, 2nd, 3rd), not the term itself.

How to help at home

• Start with times tables. Ask what 3, 6, 9, 12 has to do with 5, 8, 11, 14. Spotting the times table underneath is most of the method. The times table grid is handy to have out.
• Build a sequence together. One of you picks a rule like 4n + 1 and writes the first five terms; the other works out the rule.
• Ask "will it ever reach 100?" It shows whether your child understands what n stands for.
• Ask the school what's coming. Schools choose the order of topics across Years 7 to 9, so their scheme of work or curriculum map shows when sequences come up.

Where the nth term fits

Primary school lays the groundwork with number patterns and term-to-term rules. Key stage 3, from Year 7 to Year 9, introduces the nth term of arithmetic sequences, and schools decide when in those three years to teach it. GCSE, in Years 10 and 11, carries on with linear sequences and adds other sequences such as square and cube numbers and Fibonacci type sequences. For the other topics in these years, see the KS3 maths guide.

nth term questions to try

1. Write the first four terms of the sequence with nth term 2n + 5.
2. Find the nth term of 4, 9, 14, 19, 24.
3. Find the nth term of 1, 3, 5, 7, 9.
4. Find the nth term of 30, 26, 22, 18.
5. Is 75 a term of the sequence with nth term 4n + 3? Explain how you know.
6. What is the 50th term of 6n − 1?

Answers

1. 7, 9, 11, 13
2. 5n − 1 (the 5 times table, take away 1)
3. 2n − 1 (the 2 times table, take away 1)
4. 34 − 4n (the difference is −4, and −4 + 34 = 30)
5. Yes. 4n + 3 = 75 means 4n = 72, so n = 18. It's the 18th term.
6. 299, because 6 × 50 = 300 and 300 − 1 = 299.