Key Stage 4 maths · Algebra

GCSE sequences: special numbers and the nth term explained

At GCSE, sequences go well beyond “add the same number each time”. Pupils recognise triangular, square and cube numbers, work with arithmetic and simple geometric progressions, Fibonacci-type sequences and quadratic sequences, and learn to write an expression for the nth term of any linear sequence. It is a compact topic, but it appears as both quick marks and multi-step reasoning questions.

Special sequences and the nth term, explained

A sequence is a list of numbers that follows a rule, and there are two ways to describe that rule. A term-to-term rule says how to get from one term to the next, such as “add 4”. A position-to-term rule, usually called the nth term, gives the value of any term straight from its position, such as 4n + 3. The nth term is far more powerful: to find the 100th term you put n = 100 into the expression rather than adding 4 ninety-nine times.

An arithmetic progression goes up or down by the same amount each time, called the common difference. For a linear sequence that difference tells you the number in front of n, and you then adjust for where the sequence actually starts. Pupils also use the nth term to decide whether a number is in a sequence, and to work with sequences that go down, such as 20, 17, 14, 11, whose nth term is 23 − 3n.

Three families of number sequences are expected to be instantly recognisable. Square numbers are 1, 4, 9, 16, 25 and have nth term n². Cube numbers are 1, 8, 27, 64, 125 and have nth term n³. Triangular numbers are 1, 3, 6, 10, 15: each is made by adding the next whole number, and they can be pictured as rows of dots building a triangle. Some schools also show the formula n(n + 1) ÷ 2, which gives 210 for the 20th triangular number.

A Fibonacci-type sequence is one where every term is the sum of the two terms before it. The famous one starts 1, 1, 2, 3, 5, 8, 13, but a Fibonacci-type sequence can begin with any two numbers, for example 2, 5, 7, 12, 19. Questions often give two terms in the middle and ask pupils to work backwards, using subtraction instead of addition, to find the start.

A geometric progression multiplies by the same number each time, called the common ratio. The programme of study names sequences of the form rⁿ, where r is a positive fraction or whole number: 2ⁿ gives 2, 4, 8, 16 and (½)ⁿ gives ½, ¼, ⅛. Pupils also meet quadratic sequences, where the differences between terms are not constant but the differences between those differences are, as in 2, 5, 10, 17, 26. At this level they recognise and continue them. Writing the nth term of a general quadratic sequence belongs to the higher-tier material.

Words your child will hear

nth term: An expression that gives any term from its position n. If the nth term is 4n + 3, the 10th term is 43.
Term-to-term rule: A rule that gets each term from the one before, such as “add 4” or “multiply by 3”.
Arithmetic progression: A sequence with a constant difference between terms, such as 7, 11, 15, 19.
Common ratio: The number a geometric progression is multiplied by each time. In 3, 6, 12, 24 it is 2.
Triangular numbers: 1, 3, 6, 10, 15 and so on: the totals made by adding 1, then 2, then 3, then 4.
Fibonacci-type sequence: A sequence where each term is the sum of the two before it, starting from any two numbers.
Quadratic sequence: A sequence whose differences change by a constant amount, so the rule involves n².

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressions
Recognise and use Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rⁿ where n is an integer, and r is a positive rational number)
Deduce expressions to calculate the nth term of linear sequences

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

The first four terms of a sequence are 7, 11, 15, 19. Find the nth term, then decide whether 150 is in the sequence.

Find the common difference: 11 − 7 = 4, 15 − 11 = 4 and 19 − 15 = 4. It is the same each time, so the sequence is linear and the nth term starts with 4n.

Write the multiples of 4 under the positions and compare them with the actual terms.

n        1    2    3    4
4n       4    8   12   16
term     7   11   15   19
term-4n  3    3    3    3

Every term is 3 more than 4n, so the nth term is 4n + 3.
Check with a term you did not use. The 5th term should be 19 + 4 = 23, and 4 × 5 + 3 = 23. It works.
To test 150, solve 4n + 3 = 150. That gives 4n = 147 and n = 36.75. A position must be a whole number, so 150 is not in the sequence. (A quick check agrees: every term of 4n + 3 is odd, and 150 is even.)

The nth term is 4n + 3, and 150 is not a term because n would be 36.75.

Common mistakes and how to help

Giving just the difference as the nth term, such as “4n” or “+4”

The term-to-term rule says “add 4” and the first step of the method produces 4n, so the adjustment at the end feels like an optional extra.

What helps: Always test n = 1. Here 4 × 1 = 4 but the first term is 7, so something is missing. Substituting the first term is the fastest check there is.

Getting the signs wrong for a decreasing sequence

For 20, 17, 14, 11 the difference is −3, but pupils often write 3n + something. Then n = 1 gives a number far from 20.

What helps: Start with −3n. At n = 1 that gives −3, and 20 is 23 more, so the nth term is −3n + 23, usually written 23 − 3n. Check it gives 20, 17, 14.

Mixing up squaring and doubling, or cubing and trebling

Index notation looks like multiplication, so 4² is read as 4 × 2 = 8 and 4³ as 4 × 3 = 12.

What helps: Build them. A 4 by 4 square of counters has 16 in it; a 3 by 3 by 3 block of cubes has 27. Link n² to area and n³ to volume.

Deciding the pattern from the first three terms and being sure of the next

Seeing 2, 4, 8, most people say 16 because doubling comes to mind first. But 2, 4, 8, 14 fits equally well, with differences of 2, 4, 6.

What helps: Ask what type of sequence the question says it is, and test a proposed rule against every term given. In a reasoning question, one example never proves a rule.

How to help at home

Build the numbers with small objects. Use buttons, coins or pasta shapes to lay out square numbers (1, 4, 9) and triangular numbers (1, 3, 6). Seeing a triangle of 10 counters makes the pattern memorable in a way a list does not.
Invent a sequence and swap. Choose an nth term such as 5n − 2, write out the first four terms, and give them to your teenager to work backwards. Then swap, and let them set one for you with a deliberate mistake for you to find.
Hide two terms in a Fibonacci-type sequence. Pick any two starting numbers, write six terms, then cover two in the middle. Working out the missing terms with subtraction is exactly what exam questions ask.
Spot sequences in real plans. Saving £5 a week into an account that already holds £20 gives 25, 30, 35, which is 5n + 20. Phone tariffs, gym joining fees and taxi fares often have the same shape.
Feel geometric growth. Take a calculator, type 1 and keep multiplying by 1.5. After ten steps the number is over 57. Ask your teenager how it compares with adding 1.5 ten times.

Builds on

Key Stage 3: Algebraic notation and formulae
Key Stage 3: Solving linear equations
Key Stage 3: Graphs of linear and quadratic functions
Key Stage 3: Sequences and the nth term