Number

Number Patterns & Sequences

Spot the rule, continue a sequence, and find the nth term — arithmetic, special and quadratic sequences

Explanation & Worked Examples

Sequences & Term-to-Term Rules

A sequence is an ordered list of numbers. Each number is called a term. The term-to-term rule tells you how to get from one term to the next.

Arithmetic — add or subtract the same number each time.
3, 7, 11, 15, …  (rule: add 4)
Geometric — multiply or divide by the same number each time.
2, 6, 18, 54, …  (rule: multiply by 3)

Special Sequences to Recognise

NameSequenceRule
Square numbers1, 4, 9, 16, 25, …n × n
Cube numbers1, 8, 27, 64, …n × n × n
Triangular numbers1, 3, 6, 10, 15, …add 2, then 3, then 4 …
Fibonacci1, 1, 2, 3, 5, 8, 13, …add the two before it
How to spot the type: first check whether the gap between terms is constant (arithmetic). If the gaps grow, look at whether each term is a multiple of the last (geometric), or whether the gaps themselves form a pattern (a special or quadratic sequence).

The nth Term (Position-to-Term)

The nth term is a formula that gives any term directly from its position n (n = 1 for the first term, 2 for the second, and so on). It saves you continuing a long sequence by hand.

nth term of an arithmetic sequence

nth term = (common difference) × n + (the "zero term")

The common difference d is what you add each time; the zero term is the value before the first term (first term − d).

Sequence 5, 8, 11, 14, … → d = 3, zero term = 5 − 3 = 2 → nth term = 3n + 2
Check: n = 1 → 3(1)+2 = 5 ✓   n = 4 → 3(4)+2 = 14 ✓

Using the nth term

  • Find a term: substitute its position. 20th term of 3n + 2 = 3(20) + 2 = 62.
  • Find which term equals a value: set the formula equal to it and solve. 3n + 2 = 50 → n = 16, so 50 is the 16th term.
Quick gradient trick: the number in front of n is always the common difference. A sequence that goes up by 4 each time has an nth term starting 4n; one going down by 2 starts −2n.

Easy Examples

Continue a sequence or describe its term-to-term rule.

Example 1 — Next Term (arithmetic)

Write the next term: 3, 7, 11, 15, …

1Find the gap: 7 − 3 = 4, and it stays 4 each time (add 4)
215 + 4 = 19
Example 2 — Missing Term

Find the missing term: 4, __, 12, 16

1The gap is 4 each time (16 − 12 = 4)
24 + 4 = 8
Example 3 — Geometric Rule

Continue: 2, 6, 18, 54, …

1Each term is the last × 3 (6 ÷ 2 = 3)
254 × 3 = 162
Example 4 — A Special Sequence

Continue the triangular numbers: 1, 3, 6, 10, …

1The gaps grow by one each time: +2, +3, +4, so next is +5
210 + 5 = 15

Medium Examples

Find and use the nth term of an arithmetic sequence.

Example 1 — Find the nth Term

Find the nth term of 5, 8, 11, 14, …

1Common difference d = 3, so the formula starts 3n
2Zero term = first term − d = 5 − 3 = 2
3nth term = 3n + 2
Example 2 — Find a Specific Term

The nth term of a sequence is 4n − 1. Find the 10th term.

1Substitute n = 10: 4(10) − 1
2= 40 − 1 = 39
Example 3 — Which Term Equals a Value?

Is 50 a term of the sequence with nth term 3n − 1? If so, which one?

1Set the formula equal to 50: 3n − 1 = 50
23n = 51 → n = 17 (a whole number), so 50 is the 17th term
Example 4 — Decreasing Sequence

Find the nth term of 10, 8, 6, 4, …

1d = −2, so the formula starts −2n
2Zero term = 10 − (−2) = 12
3nth term = −2n + 12 (i.e. 12 − 2n)
Check both ends: always test your nth term with n = 1 and one later term. If both match, the formula is right.

Complex Examples

Quadratic sequences (the gaps change) and special-number formulas.

Example 1 — Recognise a Quadratic Sequence

Find the nth term of 2, 5, 10, 17, …

1First differences: 3, 5, 7 — not constant. Second differences: 2, 2 — constant, so it is quadratic
2Second difference ÷ 2 = 1, so it is based on n². Compare with 1, 4, 9, 16
3Each term is 1 more than n²: nth term = n² + 1
Example 2 — Coefficient of n²

Find the nth term of 3, 9, 19, 33, …

1First differences 6, 10, 14; second difference = 4
2Second difference ÷ 2 = 2, so start with 2n²: 2, 8, 18, 32
3Each term is 1 more: nth term = 2n² + 1
Example 3 — Triangular Number Formula

The nth triangular number is n(n + 1) ÷ 2. Find the 10th triangular number.

1Substitute n = 10: 10 × 11 ÷ 2
2= 110 ÷ 2 = 55
Example 4 — Fibonacci-style

Each term is the sum of the two before it: 5, 8, 13, 21, … Find the next term.

1Add the last two terms: 13 + 21
2= 34
The second-difference test: if the gaps between terms are not constant, look at the gaps between the gaps. A constant second difference means a quadratic sequence — and half that second difference is the number in front of n².
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Practice Questions

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