Sequences & Patterns

What Your Child Is Learning

A sequence is a list of numbers that follows a rule. Your child might see: 2, 5, 8, 11, 14... and be asked "what comes next?" The answer is 17 — because each number is 3 more than the one before it. That "3 more each time" is called the common difference.

But schools don't just want children to spot the next number. They want them to find a formula that works for any position in the sequence. "What's the 100th number?" You can't count up one by one to find that — you need the nth term rule. For the sequence above, the rule is \(3n + (-1)\), which we write as \(3n - 1\). Plug in \(n = 100\) and you get \(3 \times 100 - 1 = 299\).

This is your child's first real experience of building a general formula from a pattern. It's a stepping stone to much bigger ideas in GCSE and A-level maths, and it teaches them to look for structure rather than just answers.

How Schools Teach It Now

Schools teach a systematic method for finding the nth term of a linear sequence (one with a constant common difference). Here's how they'd find the nth term of 5, 8, 11, 14...:

1. Find the common difference. Subtract consecutive terms: \(8 - 5 = 3\), \(11 - 8 = 3\), \(14 - 11 = 3\). The common difference is 3. This becomes the number in front of \(n\).
2. Write down the \(3n\) sequence. The "\(3\) times table" gives: 3, 6, 9, 12... Compare this to our sequence: 5, 8, 11, 14...
3. Find the adjustment. Our sequence is always 2 more than the \(3n\) sequence. \(5 - 3 = 2\), \(8 - 6 = 2\), \(11 - 9 = 2\). So we add 2.
4. Write the nth term. \(n\text{th term} = 3n + 2\).
5. Check it. 1st term: \(3(1) + 2 = 5\) — correct. 2nd term: \(3(2) + 2 = 8\) — correct. 3rd term: \(3(3) + 2 = 11\) — correct.

Common Mistakes to Watch For

• Confusing the first term with the constant. In the sequence 5, 8, 11, 14..., the first term is 5 but the nth term is \(3n + 2\), not \(3n + 5\). Children often assume the "+something" in the formula must be the first term. It's not — it's the difference between the first term and the common difference. \(5 - 3 = 2\), so the constant is 2.
• Thinking all sequences are linear. Some sequences don't have a constant difference. 1, 4, 9, 16, 25... has differences of 3, 5, 7, 9 — the differences themselves change. This is a quadratic sequence (square numbers). At KS3, most sequences will be linear, but your child should check that the difference is constant before using the \(dn + c\) method.
• Getting the sign wrong on the constant. If the sequence is 1, 4, 7, 10..., the common difference is 3. The \(3n\) sequence gives 3, 6, 9, 12... Our sequence is 2 less each time, so the nth term is \(3n - 2\). Children sometimes write \(3n + 2\) because they see a "2" and default to adding.
• Forgetting to check. The quickest way to catch errors is to substitute \(n = 1\) back into the formula. If it doesn't give the first term of the sequence, something is wrong.

Try It Together

Try another one together: find the nth term of 4, 7, 10, 13... (Answer: common difference = 3, \(4 - 3 = 1\), so nth term = \(3n + 1\). Check: \(3(1) + 1 = 4\).)

Let Them Practise

This free game lets your child practise finding patterns and rules in number sequences.

MaffsGames needs no account and no sign-up; what it records while your child plays is set out on the privacy page.