Sequences
Master sequences for GCSE Maths with structured, exam-style practice. This Foundation resource covers continuing and describing number sequences and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Find the rule by looking at how you get from one term to the next.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Topic overview
A sequence is an ordered list of numbers following a rule. Each number is a term, and the position of a term is often written as \(n\), so the first term is \(n = 1\).
Most GCSE sequences are linear, meaning you add or subtract the same amount each time. That fixed amount is the common difference, and finding it is usually the first thing to do.
Other sequence types appear too. Geometric sequences multiply by a fixed number, and special sequences such as the square numbers, cube numbers, triangular numbers and the Fibonacci sequence are worth recognising on sight, because questions often use them without naming them.
Revision notes
Finding the rule
Work out the difference between consecutive terms. If it is constant, the sequence is linear.
For \(4, 7, 10, 13\), the difference is \(+3\) each time, so the rule is add \(3\). Continuing gives \(16\) and \(19\).
Special sequences to recognise
Square numbers: \(1, 4, 9, 16, 25\). Cube numbers: \(1, 8, 27, 64\). Triangular numbers: \(1, 3, 6, 10, 15\).
The Fibonacci sequence adds the previous two terms: \(1, 1, 2, 3, 5, 8, 13\). Recognising these saves working out a rule from scratch.
Term-to-term rules
A term-to-term rule tells you how to get from one term to the next, and needs a starting value to be complete.
So start at \(5\) and add \(4\) gives \(5, 9, 13, 17\). This differs from a position-to-term rule, which lets you jump straight to any term without listing the ones before it.
Key points
- A sequence is an ordered list following a rule.
- Each number in a sequence is a term.
- A linear sequence has a constant difference.
- A geometric sequence multiplies by a fixed number.
- A term-to-term rule needs a starting value.
- Learn the square, cube, triangular and Fibonacci sequences.
Worked examples
Example 1
Write down the next two terms of \(5, 9, 13, 17, \ldots\)
Working
Example 2
Find the next term in \(1, 3, 6, 10, \ldots\)
Working
Example 3
A sequence starts at \(3\) and the rule is multiply by \(2\). Write the first four terms.
Working
Common mistakes
Assuming every sequence is linear.
Check the differences first. If they are not constant, the sequence may be quadratic, geometric or special.
Giving a term-to-term rule without a starting value.
Add 4 describes infinitely many sequences. The first term is needed too.
Confusing the term with its position.
In 5, 9, 13, the third term is 13, not 3.
Miscounting the differences.
Work them out one at a time and write them between the terms.
Exam tips
- Write the differences between the terms above the gaps.
- Check whether the sequence is one of the special ones before hunting for a rule.
- Give both the starting value and the rule when describing a sequence.
- Check your rule works for every term given, not just the first two.
Key terms
- Term
- A single number in a sequence.
- Common difference
- The fixed amount added each time in a linear sequence.
- Term-to-term rule
- A rule giving the next term from the current one.
- Triangular number
- A number in the sequence 1, 3, 6, 10, 15.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.