Continue and describe sequences, find the nth term of a linear (arithmetic) sequence, and recognise quadratic, geometric, Fibonacci and other special sequences.

What this dot point is asking

WJEC asks you to continue and describe number sequences, to find the nth term (the position-to-term rule) of a linear sequence, and to recognise special sequences such as square, cube, triangular, geometric and Fibonacci-type. The nth term is the key algebraic skill, because it lets you find any term directly and decide whether a given number belongs to the sequence. It appears on both components, and the reasoning involved (justifying whether a value is a term) is exactly the AO2 thinking WJEC rewards.

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

There are two ways to describe a sequence. A term-to-term rule tells you how to move from one term to the next, such as "add $4$". It is easy to state but you must work through every term to reach a far-off one. A position-to-term rule, the nth term, gives any term directly from its position number $n$, which is far more powerful.

Finding the nth term of a linear sequence

A linear sequence has a constant difference between consecutive terms.

Deciding whether a number is in the sequence

To test membership, set the nth term equal to the number and solve for $n$. If $n$ comes out as a positive whole number, the value is a term in that position; if $n$ is a fraction or negative, it is not. This is a favourite WJEC reasoning question, and the marks come from the algebra and the justified conclusion, not from a bare yes or no.

Special sequences

Beyond linear sequences, WJEC expects you to recognise several patterns: the square numbers $1, 4, 9, 16, \ldots$ (nth term $n^2$), the cube numbers $1, 8, 27, \ldots$ (nth term $n^3$), the triangular numbers $1, 3, 6, 10, \ldots$, and Fibonacci-type sequences where each term is the sum of the previous two ($1, 1, 2, 3, 5, 8, \ldots$). A geometric sequence multiplies by a constant ratio each time, such as $3, 6, 12, 24, \ldots$ (doubling). At Higher tier you may be asked for the nth term of a simple quadratic sequence, where the second differences are constant.

Spotting which type a sequence is

A quick diagnostic separates the types. Find the differences between consecutive terms: if they are constant, the sequence is linear and you use $dn + c$. If the first differences change but the second differences are constant, it is a quadratic sequence whose nth term involves $n^2$. If each term is a fixed multiple of the previous one, it is geometric. If neither differences nor ratios are constant but each term is the sum of the two before, it is Fibonacci-type. Running this check before writing anything prevents trying to force a linear rule onto a non-linear pattern.

Patterns from diagrams

Many WJEC sequence questions present a growing pattern of dots, matchsticks or tiles and ask for the number in the nth diagram. Count the items in the first few patterns to form a number sequence, then find its nth term as usual. The advantage of the diagram is that it often explains why the rule works, for example a row of $n$ squares needing $3n + 1$ matchsticks because each new square adds three sticks to a starting one. Linking the picture to the formula is exactly the reasoning WJEC rewards, so describe the rule in words as well as algebra where asked.

Why this matters

Sequences link arithmetic and algebra: finding an nth term is essentially building a formula from a pattern, the same modelling skill used in straight line graphs (a linear sequence plotted against position gives a straight line of gradient $d$). The membership-test questions are a clean source of reasoning marks, and recognising special sequences supports number work elsewhere. WJEC rewards showing the difference, the adjustment and the conclusion in full.