The nth term, explained with examples

By The STEM-Tutor team · Updated October 2026

The nth term is a rule that gives any term in a sequence from its position. Instead of counting along, you can jump straight to the 100th term.

What the nth term means

A sequence is a list of numbers that follows a pattern, such as 5, 8, 11, 14, …. The position of a term is n: the first term has n = 1, the second n = 2, and so on. The nth term is a formula in n that produces every term. For 5, 8, 11, 14, … the nth term is 3n + 2. Check: when n = 1, 3 + 2 = 5; when n = 2, 6 + 2 = 8. It works.

A term-to-term rule, such as “add 3”, tells you how to get from one term to the next. A position-to-term rule, the nth term, tells you any term directly. The nth term is more powerful because you do not need the terms before it.

Linear sequences

A linear (or arithmetic) sequence goes up or down by the same amount each time. That amount is the common difference, d. The nth term always has the form dn + c.

1. Find the common difference d. This is the number in front of n.
2. Write the sequence dn: the d times table.
3. Compare it with the actual sequence. The adjustment from dn to the sequence is c.

Worked examples

Example 1: an increasing sequence

Find the nth term of 7, 11, 15, 19, …

• The difference is 4, so start with 4n: 4, 8, 12, 16.
• Each term is 3 more than 4n, so the nth term is 4n + 3.

Example 2: a decreasing sequence

Find the nth term of 20, 17, 14, 11, …

• The difference is −3, so start with −3n: −3, −6, −9, −12.
• Each term is 23 more, so the nth term is 23 − 3n, which can also be written −3n + 23.

Example 3: finding a particular term

Find the 50th term of the sequence with nth term 4n + 3.

• Substitute n = 50: 4 × 50 + 3 = 203.

Example 4: is a number in the sequence?

Is 100 a term in the sequence 4n + 3?

• Set 4n + 3 = 100, so 4n = 97 and n = 24.25.
• n must be a whole number, so 100 is not in the sequence. This is a common exam question and the reasoning must be written down.

Example 5: the first term over a value

Find the first term of 4n + 3 that is greater than 500.

• 4n + 3 > 500 gives n > 124.25, so n = 125.
• The term is 4 × 125 + 3 = 503.

Quadratic sequences (Higher)

In a quadratic sequence, the first differences are not constant, but the second differences are. The nth term has the form an² + bn + c, and a is half the second difference.

1. Find the first differences, then the second differences.
2. Halve the second difference to get a.
3. Subtract an² from each term. What is left is a linear sequence.
4. Find the nth term of that linear sequence and add it to an².

Example 6: a quadratic sequence

Find the nth term of 4, 9, 16, 25, 36, …

• First differences: 5, 7, 9, 11. Second differences: 2, so a = 1.
• Subtract n² (1, 4, 9, 16, 25): 3, 5, 7, 9, 11, a linear sequence with nth term 2n + 1.
• The nth term is n² + 2n + 1, which is (n + 1)², as you might have spotted.

Example 7: a harder quadratic

Find the nth term of 3, 10, 21, 36, 55, …

• First differences: 7, 11, 15, 19. Second differences: 4, so a = 2.
• Subtract 2n² (2, 8, 18, 32, 50): 1, 2, 3, 4, 5, which is n.
• The nth term is 2n² + n. Check n = 3: 18 + 3 = 21.

Other sequences to recognise

• Square numbers: 1, 4, 9, 16, … with nth term n².
• Cube numbers: 1, 8, 27, 64, … with nth term n³.
• Triangular numbers: 1, 3, 6, 10, … with nth term n(n + 1)/2.
• Geometric sequences: each term is multiplied by the same number, the common ratio r. For 3, 6, 12, 24, … r = 2 and the nth term is 3 × 2^{n − 1}.
• Fibonacci-type sequences: each term is the sum of the two before, such as 1, 1, 2, 3, 5, 8, ….

To identify a sequence, check differences first. Constant first differences mean linear; constant second differences mean quadratic. If the ratio between terms is constant, it is geometric.

Sequences from patterns

Exam questions often show a pattern of shapes, such as matchsticks or tiles, and ask for the number of pieces in pattern n. Count the pieces in the first few patterns, write them as a sequence, then find the nth term in the usual way. Finally, use your formula to answer the question, for example how many matchsticks are needed for pattern 20, or which pattern uses exactly 61 matchsticks.

Example 8: a matchstick pattern

Pattern 1 is a square made of 4 matchsticks. Each new pattern adds one square to the right, sharing a side. How many matchsticks are in pattern n, and which pattern uses 61?

• The counts are 4, 7, 10, 13, …, going up by 3 each time.
• The nth term is 3n + 1. Setting 3n + 1 = 61 gives n = 20, so pattern 20 uses 61 matchsticks.

Common mistakes

• Using the first term instead of the common difference as the number in front of n.
• Getting the sign wrong for decreasing sequences.
• Saying a number is in the sequence without checking that n is a whole number.
• Using the first differences instead of half the second difference for a.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is the nth term on the Foundation paper?

Yes, for linear sequences. Quadratic sequences are on the Higher tier.

What is the difference between arithmetic and linear sequences?

They are the same thing: a sequence with a constant difference. Arithmetic is the term used at A level, where you also find the sum of an arithmetic series.

Key takeaways

• The nth term gives any term from its position n.
• Linear sequences: the common difference goes in front of n.
• Check if a number is in a sequence by solving for n.
• Quadratic sequences: a is half the second difference.
• Recognise square, cube, triangular, geometric and Fibonacci-type sequences.

Related guides

• GCSE Maths topics
• Factorising quadratics
• GCSE Maths revision