Sequences
Arithmetic, quadratic, geometric and Fibonacci sequences and the nth term.
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Every point below, worked through in one video, and each point links straight to its own part.
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Sequences, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Continue a sequence and describe its term-to-term rule
The term-to-term rule says how to get from one term to the next: 5, 8, 11, 14 has the rule "add 3". Check the rule holds across SEVERAL gaps, not just the first one - some sequences change by a different amount each step.
Recognise arithmetic, geometric, square, cube, triangular and Fibonacci-type sequences
Arithmetic sequences add the same number each time; geometric ones multiply by the same number (2, 6, 18, 54). Squares (1, 4, 9, 16), cubes (1, 8, 27, 64) and triangular numbers (1, 3, 6, 10) come straight from their shapes, and Fibonacci-type sequences add the previous two terms.
Find and use the nth term of an arithmetic sequence
The common difference is the coefficient of n. For 7, 11, 15, 19 the difference is 4, so start from 4n; the sequence sits 3 above the 4 times table, giving 4n + 3. Now any term is a substitution away: the 50th term is 4 × 50 + 3 = 203.
Decide whether a given number is a term of a sequence
Set the nth term equal to the number and solve. Is 150 in the sequence 4n + 3? Then 4n + 3 = 150 gives n = 36.75 - not a whole number, so no. A whole-number n means yes, and tells you which term it is.
Find the nth term of a quadratic sequenceHigher only
If the first differences keep changing but the SECOND differences are constant, the sequence is quadratic, and the n² coefficient is half that second difference. For 3, 8, 15, 24 (second difference 2): start with n², subtract it from each term, and find the nth term of what remains - here n² + 2n.
Common mistakes in Sequences
The slips that cost marks year after year, and what to do instead.
Using the difference as the number added on, so 7, 11, 15, 19 gets the nth term n + 4.
The difference goes in front of n. Start with 4n, which gives 4 for the first term; you need 7, so add 3 to get 4n + 3.
Using a positive difference for a sequence that goes down, so 20, 17, 14, 11 becomes 3n + 17.
The terms fall by 3, so start from −3n. That gives −3 for the first term and you need 20, so add 23: the nth term is 23 − 3n.
Rounding n to decide whether a number is in a sequence, so 4n + 3 = 150 is read as the 37th term.
Only a whole-number n puts the number in the sequence. Here 4n = 147 gives n = 36.75, so 150 is not a term.
Giving the term-to-term rule, like "add 4", when the question asks for the nth term.
The nth term is a rule in n that finds any term directly, such as 4n + 3. "Add 4" only takes you from one term to the next.
Using the whole second difference as the number in front of n² in a quadratic sequence.Higher only
Halve it. For 3, 8, 15, 24 the second difference is 2, so start with n²; taking n² away leaves 2, 4, 6, 8, which is 2n, so the nth term is n² + 2n.
Practice
Try a Sequences question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Answer: 4n + 1
The common difference is 4, so the nth term starts 4n; since 4 × 1 = 4 and the first term is 5, add 1.
- The common difference is 9 − 5 = 4, so the nth term begins 4n.
- At n = 1, 4n gives 4, but the first term is 5.
- Adjust by adding 1: nth term = 4n + 1.
- Check: 4 × 2 + 1 = 9 and 4 × 3 + 1 = 13, which match.
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