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

The Eduqas algebra content asks you to generate the terms of a sequence from a rule, find the nth term of a linear (arithmetic) sequence, and at Higher tier find the nth term of a quadratic sequence. You should also recognise common sequences on sight: arithmetic, geometric, square, cube, triangular and Fibonacci. The nth term is the key tool, because it lets you find any term directly and test whether a given number belongs to the sequence. Linear nth term appears at both tiers; the quadratic nth term is a reliable Higher-tier question.

Generating a sequence

A sequence can be defined by a position-to-term rule (the nth term) or a term-to-term rule (how to get from one term to the next). To generate from an nth term, substitute $n = 1, 2, 3, \ldots$: the rule $3n - 2$ gives $1, 4, 7, 10, \ldots$. To generate from a term-to-term rule, start with the first term and apply the rule repeatedly: "start at $5$, double each time" gives $5, 10, 20, 40, \ldots$.

The nth term of a linear sequence

A linear sequence increases by the same amount each time, the common difference $d$.

For $5, 8, 11, 14, \ldots$, the difference is $3$, so the rule starts $3n$. The $3n$ values are $3, 6, 9, 12$, which are $2$ short of the sequence, so $c = 2$ and the nth term is $3n + 2$. To test whether $50$ is a term, solve $3n + 2 = 50$: this gives $n = 16$, a whole number, so $50$ is the 16th term. If $n$ were not a whole number, the value would not be in the sequence.

The nth term of a quadratic sequence (Higher)

A quadratic sequence has a constant second difference (the differences of the differences).

Recognising special sequences

Some sequences should be known by sight. The square numbers are $1, 4, 9, 16, \ldots$ (nth term $n^2$); the cube numbers are $1, 8, 27, 64, \ldots$ (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 previous two terms. A geometric sequence multiplies by a constant ratio each time, such as $2, 6, 18, 54, \ldots$ (ratio $3$). Spotting which type a sequence is points you straight to the right method.

Using the nth term

The power of the nth term is that it gives any term without listing all the earlier ones. To find the 100th term of $3, 7, 11, \ldots$ with nth term $4n - 1$, substitute $n = 100$ to get $399$, with no need to write out the sequence. The nth term also tests membership: to ask whether $250$ is a term of $4n - 1$, solve $4n - 1 = 250$, which gives $n = 62.75$. Because $n$ is not a whole number, $250$ is not in the sequence. This "is it a term" question is a favourite Eduqas reasoning task because it rewards understanding that $n$ must be a positive integer.

Why sequences matter

Sequences link algebra to pattern-spotting and to real growth. A linear sequence models a quantity changing by a fixed amount each step (savings growing by the same deposit), while a geometric sequence models repeated proportional change (a population multiplying by a fixed factor), which connects directly to the growth and decay work in the ratio area. Finding and justifying an nth term is also an AO2 communication skill: Eduqas wants the rule stated clearly in terms of $n$, not just the next few terms, so always give the general expression when one is asked for.