Sequences and the nth term · GCSE Maths

Term-to-term rules

A sequence is a list of numbers in a pattern. The term-to-term rule tells you how to get from one term to the next: 7, 11, 15, 19 … add 4.
A geometric sequence multiplies by the same number (the common ratio) each time: 3, 6, 12, 24 … × 2.

In an arithmetic (linear) sequence, the difference between terms is the same.
nth term = difference × n + (first term − difference). For 5, 8, 11, 14 the difference is 3, so the nth term is 3n + 2.
For a decreasing sequence the difference is negative: 20, 17, 14, 11 has nth term 23 − 3n.
Substitute to find any term: the 10th term of 4n − 1 is 39.

Set the nth term equal to the number and solve. If n is a whole number, it is in the sequence.
3n + 2 = 51 gives n = 49/3, which is not a whole number, so 51 is not a term.

Square numbers: 1, 4, 9, 16…; cube numbers: 1, 8, 27, 64…; triangular numbers: 1, 3, 6, 10, 15…
Fibonacci-type sequences add the two previous terms: 1, 1, 2, 3, 5, 8, 13…
(Higher) Quadratic sequences have a constant second difference. The coefficient of n² is half the second difference: 2, 5, 10, 17… is n² + 1.

Sequence: A list of numbers that follow a rule.
Term: A number in a sequence.
Term-to-term rule: The rule for getting from one term to the next.
nth term: A formula that gives any term from its position n.
Arithmetic sequence: A sequence with a constant difference between terms.
Geometric sequence: A sequence where each term is multiplied by the same number.
Common ratio: The number each term is multiplied by in a geometric sequence.
Quadratic sequence: A sequence whose nth term includes n², with a constant second difference.