Sequences: generating terms, finding the nth term of a linear (arithmetic) sequence and a quadratic sequence (Higher tier), and recognising geometric, triangular, square, cube and Fibonacci sequences.

What this dot point is asking

A sequence is an ordered list of numbers that follow a rule. Edexcel expects you to generate terms from a rule, find the nth term of a linear (arithmetic) sequence, find the nth term of a quadratic sequence at Higher tier, and recognise special sequences such as geometric, triangular, square, cube and Fibonacci. The nth term is powerful because it lets you find any term, however far along, without listing them all.

Linear (arithmetic) sequences

A linear sequence goes up or down by the same amount each time. That constant is the first difference.

For $7, 12, 17, 22$, the difference is $5$, so the nth term is $5n + c$. Since $5 \times 1 = 5$ and the first term is $7$, you add $2$, giving $5n + 2$. A negative difference works the same way: for $20, 17, 14, 11$ the nth term is $-3n + 23$.

The nth term lets you answer "is $103$ a term?" Set $5n + 2 = 103$, solve to get $n = 20.2$; because $n$ is not a whole number, $103$ is not in the sequence.

Quadratic sequences (Higher)

A quadratic sequence has a first difference that itself changes, but the second difference is constant.

Special sequences to recognise

Some sequences appear so often that you should know them on sight: square numbers $1, 4, 9, 16, \ldots$ (nth term $n^2$); cube numbers $1, 8, 27, 64, \ldots$ (nth term $n^3$); triangular numbers $1, 3, 6, 10, 15, \ldots$ (nth term $\tfrac{n(n+1)}{2}$); and the Fibonacci sequence $1, 1, 2, 3, 5, 8, \ldots$, where each term is the sum of the two before it.

Geometric sequences

A geometric sequence multiplies by a constant ratio rather than adding. For $2, 6, 18, 54$ the ratio is $3$, so each term is three times the previous one. Geometric sequences can include fractional or negative ratios, such as $80, 40, 20, 10$ with ratio $\tfrac{1}{2}$, and surd ratios appear at Higher tier. A negative ratio makes the terms alternate in sign, as in $3, -6, 12, -24$ with ratio $-2$.

Using the nth term

The nth term is most powerful for answering questions without listing terms. To find the $50$th term of $5n + 2$, substitute $n = 50$: $5(50) + 2 = 252$. To test whether a value belongs to a sequence, set the nth term equal to it and solve: for $5n + 2 = 87$, you get $n = 17$, a whole number, so $87$ is the $17$th term; but $5n + 2 = 90$ gives $n = 17.6$, which is not a whole number, so $90$ is not in the sequence. This "is it a term?" reasoning is a frequent exam question and is only quick once you have the nth term.

Try this

Q1. Find the nth term of the linear sequence $9, 13, 17, 21$. [2 marks]

• Cue. The difference is $4$, so it starts with $4n$. Since $4 \times 1 = 4$ and the first term is $9$, add $5$: the nth term is $4n + 5$.

Q2. Find the $10$th term of a sequence with nth term $n^2 - 1$. [1 mark]

• Cue. Substitute $n = 10$: $10^2 - 1 = 99$.