Key Stage 3 maths · Algebra

KS3 sequences and the nth term: a guide for parents

A sequence is a list of numbers that follows a rule. In Key Stage 3 children generate terms from two kinds of rule, recognise arithmetic sequences and write a formula for the nth term, which lets them find any term without listing the ones before it, and meet geometric sequences and other famous patterns.

Sequences and the nth term, explained

Children in primary school continue sequences such as 3, 7, 11, 15 by spotting that each term is 4 more than the one before. That is a term-to-term rule, and it is easy to use but slow if you want the 100th term, because you would have to work through all 99 before it. At Key Stage 3 they also use a position-to-term rule, which links a term to its position number n. For a rule such as “multiply the position by 4 and add 3”, the 1st term is 4 × 1 + 3 = 7, the 2nd is 4 × 2 + 3 = 11, and the 50th is 4 × 50 + 3 = 203, found directly.

A sequence with a constant difference between terms is an arithmetic sequence. Its nth term is always the common difference multiplied by n, with a fixed number added or subtracted to make the first term come out right. For 7, 11, 15, 19 the difference is 4, so the nth term starts with 4n. When n = 1, 4n gives 4, but the first term is 7, so 3 must be added: the nth term is 4n + 3. Sequences that go down have a negative difference, so 20, 17, 14, 11 has nth term −3n + 23, often written 23 − 3n. Children check the formula by substituting n = 1, 2 and 3 and comparing with the terms they started with.

Once they have the nth term, children can answer questions without listing: what is the 80th term, or is 99 in the sequence? They solve an equation to find out. For 4n + 3 = 99 they get 4n = 96, so n = 24, a whole number, so 99 is the 24th term. For 4n + 3 = 100 they get n = 24.25, which is not a whole number, so 100 is not in the sequence. Shape and pattern problems, such as matchsticks in a row of squares, are often used to show where a sequence’s rule comes from.

Children also learn that not every sequence is arithmetic. In a geometric sequence each term is found by multiplying by a constant, called the common ratio: 3, 6, 12, 24 doubles each time, and 80, 40, 20, 10 halves. They also meet other sequences that arise in maths. Schools choose which ones, and common examples are the square numbers (1, 4, 9, 16, with nth term n²), the triangular numbers (1, 3, 6, 10) and sequences in which each term is made by adding the two before it, like 1, 1, 2, 3, 5, 8. The programme of study asks children to “appreciate” sequences like these, so the focus is on recognising and continuing them. Working out nth-term formulae for non-linear sequences such as quadratic ones comes later, at GCSE.

Words your child will hear

Term: One number in a sequence. The 3rd term of 7, 11, 15 is 15.
Term-to-term rule: A rule for getting the next term from the one before, such as “add 4”.
Position-to-term rule: A rule that gives any term directly from its position number, such as “multiply by 4 then add 3”.
nth term: A formula giving the term in position n, such as 4n + 3.
Arithmetic sequence: A sequence that goes up or down by the same amount each time.
Common difference: The constant amount added (or taken away) in an arithmetic sequence.
Geometric sequence: A sequence where each term is the previous term multiplied by the same number.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Generate terms of a sequence from either a term-to-term or a position-to-term rule
Recognise arithmetic sequences and find the nth term
Recognise geometric sequences and appreciate other sequences that arise

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

A worked example

Find the nth term of the sequence 7, 11, 15, 19, … and use it to decide whether 99 and 100 are terms in the sequence.

Find the difference between neighbouring terms.
```
7 → 11 → 15 → 19
   +4   +4   +4
```
The sequence is arithmetic with common difference 4, so the nth term begins with 4n. Write the multiples of 4 underneath and compare.
```
n:       1   2   3   4
4n:      4   8  12  16
term:    7  11  15  19
```
Each term is 3 more than 4n, so the nth term is 4n + 3. Check with n = 3: 4 × 3 + 3 = 15, which matches.
Is 99 a term? Solve 4n + 3 = 99. Subtracting 3 gives 4n = 96, so n = 24. This is a whole number, so 99 is the 24th term.
Is 100 a term? Solve 4n + 3 = 100. Subtracting 3 gives 4n = 97, so n = 24.25. A position must be a whole number, so 100 is not in the sequence.

The nth term is 4n + 3. 99 is the 24th term, but 100 is not a term.

Common mistakes and how to help

Answering “n + 4” when asked for the nth term of a sequence that goes up in 4s

Children give the term-to-term rule (“add 4”) and write it as if it were a formula. But n is the position number, not the previous term.

What helps: Test the formula. If the sequence is 7, 11, 15 and the answer is n + 4, then n = 1 gives 5, not 7. The coefficient of n is the difference, and the 1st, 2nd and 3rd terms are the check.

Forgetting the adjustment or getting its sign wrong

Children write 4n for 7, 11, 15, 19 and stop, or write 4n − 3 instead of 4n + 3. For 2, 5, 8, 11 the answer 3n − 1 needs a subtraction, which feels the wrong way round.

What helps: Write the multiples of the difference in a row above the sequence and look at what has to be added or subtracted to turn one into the other, then confirm with n = 1.

Treating a geometric sequence as arithmetic

For 2, 6, 18 a child notices the first jump is +4 and continues 22, 26. The jumps are not equal, because each term is 3 times the one before.

What helps: Check whether the difference is constant. If not, divide each term by the one before it. If the answers match, it’s geometric and the next terms are 54, 162.

Doubling the position to double the term

The 10th term of 4n + 3 is 43, so a child decides the 20th must be 86. But the +3 is added only once, so the 20th term is 4 × 20 + 3 = 83.

What helps: Always put the position into the formula. The 4n part does double when n doubles (40 becomes 80), but the +3 stays the same.

How to help at home

Build a matchstick pattern. Lay out matchsticks, or cut-up straws or pencils, to make a row of joined squares: 4, 7, 10 sticks and so on. Predict how many are needed for 10 squares, then build it to check (the answer is 31). Ask what the rule is in words, and then in letters.
Use the sequences in everyday life. Savings of £5 a week, bus routes that run every 12 minutes or a phone plan that adds a flat amount each month are all arithmetic sequences. Ask what the nth week’s total would be.
Spot the other sequences. Look for square numbers in tile patterns, triangular numbers in a rack of bowling pins or stacked cans, and doubling in a folded piece of paper (1, 2, 4, 8 layers). A pattern you can touch is easier to describe.
Always test the formula. After your child finds an nth term, ask them to check it with the first three terms. If it gives 7, 11 and 15 it works, and the habit of testing a formula pays off throughout GCSE.

Builds on

Year 6: Introduction to algebra

Leads on to

Key Stage 4: Factorising, indices and identities
Key Stage 4: Straight-line and quadratic graphs
Key Stage 4: Quadratics, simultaneous equations, inequalities
Key Stage 4: Special sequences and the nth term
Key Stage 4: Algebraic fractions, proof and functions
Key Stage 4: Exponential, trig and circle graphs
Key Stage 4: Quadratic formula, iteration, inequalities
Key Stage 4: Quadratic sequences and surd ratios