Continue sequences using a term-to-term rule, find and use the nth term of a linear sequence, recognise quadratic and special sequences, and find the nth term of a quadratic sequence (Higher tier).

What this dot point is asking

Sequences are ordered lists of numbers built by a rule, and the CCEA Algebra strand asks you to continue them and to capture the rule algebraically. You must continue a sequence using a term-to-term rule, find and use the nth term (position-to-term rule) of a linear sequence, recognise special sequences such as square, cube and triangular numbers, and at Higher tier find the nth term of a quadratic sequence using second differences. The nth term lets you jump to any term without listing them all, which is the key exam skill.

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

A term-to-term rule tells you how each term follows from the previous one, such as "add 3" or "multiply by 2". It is easy to continue a sequence with, but slow if you want a distant term.

A position-to-term rule, the nth term, gives the value at any position $n$ directly. For the sequence $4, 7, 10, 13, \ldots$ the nth term is $3n + 1$, so the 100th term is $3 \times 100 + 1 = 301$ without listing the terms in between. This is why the nth term is the powerful tool.

The nth term of a linear sequence

A linear (arithmetic) sequence goes up or down by the same amount each time, the common difference. The nth term always begins with the common difference times $n$.

So for $9, 14, 19, 24, \ldots$ the difference is $5$, $5n$ gives $5, 10, 15, 20$, and each term is $4$ more, so the nth term is $5n + 4$. A negative common difference works the same way: for $20, 17, 14, \ldots$ the nth term is $-3n + 23$.

Special sequences

Some sequences are worth recognising on sight. The square numbers are $1, 4, 9, 16, \ldots$ with nth term $n^2$; the cube numbers are $1, 8, 27, 64, \ldots$ with nth term $n^3$; the triangular numbers are $1, 3, 6, 10, 15, \ldots$ with nth term $\tfrac{n(n+1)}{2}$. The Fibonacci sequence $1, 1, 2, 3, 5, 8, \ldots$ adds the two previous terms. Spotting these speeds up many questions.

Quadratic sequences (Higher)

A quadratic sequence does not have a constant first difference, but its second differences are constant.

Why this matters

Sequences train you to spot pattern and structure and to express it algebraically, which is exactly the reasoning CCEA wants. The linear nth term connects to straight-line graphs (the common difference is the gradient), and quadratic sequences connect to quadratic graphs and expansion. Recognising the special number sequences also pays off in number and geometry questions.