Generate sequences from a rule; find the nth term of a linear sequence and a quadratic sequence (Higher tier); and recognise arithmetic, geometric, square, cube, triangular and Fibonacci sequences.

What this dot point is asking

OCR reference A5 asks you to generate sequences from a term-to-term or position-to-term rule, to find the nth term of a linear sequence (and a quadratic sequence at Higher tier), and to recognise special sequences such as arithmetic, geometric, square, cube, triangular and Fibonacci. The nth term is the engine: it lets you find any term directly and test whether a value belongs. Sequences appear on every paper and link to straight-line and quadratic graphs.

Linear sequences and the nth term

A linear sequence goes up (or down) by the same amount each time.

For $4, 7, 10, 13$ the difference is $3$, so the rule starts $3n$; since $3 \times 1 = 3$ but the first term is $4$, add $1$ to get $3n + 1$. A decreasing sequence has a negative difference: $20, 17, 14, 11$ has nth term $-3n + 23$. The nth term answers both "what is the $100$th term?" (substitute $n = 100$) and "is $61$ in the sequence?" (solve $3n + 1 = 61$ and check $n$ is a whole number).

Recognising special sequences

Several named sequences recur in exams.

The square numbers are $1, 4, 9, 16, 25$ (nth term $n^2$); the cube numbers are $1, 8, 27, 64$ (nth term $n^3$); the triangular numbers are $1, 3, 6, 10, 15$ (nth term $\tfrac{n(n+1)}{2}$). A Fibonacci-type sequence adds the previous two terms, as in $2, 3, 5, 8, 13$. A geometric sequence multiplies by a constant ratio, as in $3, 6, 12, 24$ (ratio $2$). Spotting these quickly often unlocks a question that looks harder than it is.

OCR sometimes describes a sequence by a position-to-term rule (substitute the position $n$ to get the term) or a term-to-term rule (do something to the previous term). A position-to-term rule such as $n^2 + 1$ lets you jump straight to the $20$th term, whereas a term-to-term rule such as "double and add one" forces you to work through each term in turn. Being asked to convert between the two descriptions, or to continue a sequence given one of them, is common, so be comfortable reading both.

Quadratic sequences (Higher)

A quadratic sequence has a changing first difference but a constant second difference.

Why the nth term matters

The nth term turns a list into a formula, which is exactly the position-to-term thinking OCR wants. It connects to graphs: a linear sequence plotted against $n$ gives points on a straight line, and a quadratic sequence gives points on a parabola. Forming and using the nth term is also an AO2 reasoning skill, so showing the difference table and the adjustment step secures method marks even if the final expression has a slip.