Sequences and Nth Term (Linear) – GCSE Maths Revision and Worksheets

If you are revising linear sequences or how to find the nth term for a sequence then the following resources and revision materials should be very helpful. From basic sequence questions to complex nth term and applied questions, the GCSE Maths sequence revision materials will help to stretch all abilities of candidate and are suitable for all major exam boards including AQA and OCR as well as AQA. If you are looking to revise quadratic sequences then visit our dedicated page.

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What you need to know

What you need to know:

In GCSE maths, a sequence (sometimes called a progression) is an ordered list of numbers that follow a pattern or a rule. Two important examples are: arithmetic sequences, where each next term is calculated by adding a fixed number to the previous term, and geometric sequences, where each next term is calculated by multiplying the previous term by a fixed term.

There are two primary ways that we can define the rule of a sequence, and they are:

  • Term-to-term: each term in the sequence is calculated by performing a fixed set of operations (such as multiply by 2, then subtract 6) to the previous term;
  • Position-to-term: each term in any sequence is numbered according to its position (the first term is number 1, the fifth term is number 5), and in this method the value of each term is calculated by performing a fixed set of operations to its own position number.

A question may give you the rule for a sequence (and it could be term-to-term or position-to-term) and then ask you to calculate several terms in the sequence.

It is equally possible that a question gives you a several terms of a sequence that have already been calculated, and then asks you questions about it. You should be able to recognise some simple, familiar sequences such as: square and cube numbers, triangular numbers, and simple arithmetic progressions. Other, slightly trickier sequences you may also be required to recognise are: sequences that follow a similar pattern to the Fibonacci sequence (add together the previous two terms to get the next term), quadratic sequences (ones where the position-to-term rule for them is a quadratic), and simple geometric sequences. In the foundation course, geometric sequences will always involve multiplying by a rational number (either a whole number or a fraction) at each step, however in the higher course it could be that the rule involves multiplying by a surd.

Finally, it is also possible that you are asked the work out the formula for the nth term of a linear sequence (which, in familiar terms, will be the position-to-term rule for an arithmetic sequence). For example, we might work out that formula for the nth term of the sequence 3, 7, 11, 15, 19, … is given by u_n = 4n - 1, where u_n is used (as it often is) to denote the value of the term that has position number n, a.k.a. the nth term. Calculating the 4th term, we get: u_4 = (4 \times 4) - 1 = 15, which agrees with our original sequence.

Linear sequences and finding the nth term are topics that students often make silly errors on. These errors can certainly be reduced by getting them to practise different questions and exam style problems. At Maths Made Easy we have made the effort to source what we think are the best GCSE Maths linear sequence revision resources. We hope tutors, teachers and students find these useful.

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