11+ Sequences: How to Find the Rule and the nth Term Without Guessing

Sequence questions look like the easiest marks on an 11+ maths paper. Four numbers, find the next one. Then the question asks for the 50th term, or for two missing numbers in the middle, and children who have been spotting patterns by eye start guessing.

The good news is that there is a method, and it is short. This post walks through it one step at a time, with the questions I use to check whether a child has it.

What 11+ sequence questions ask

Most sequence questions are one of five kinds.

• Next term, the simplest, where your child continues the pattern.
• Missing terms, where one or more numbers in the middle have been taken out.
• The rule in words, such as "start at 5 and add 3 each time".
• A far away term, such as the 20th, 50th or 100th, which is too far to count.
• Is this number in the sequence? For example, does 50 appear in 5, 8, 11, 14 and so on?

Pattern questions with shapes, such as rows of squares made from matchsticks, are the same questions in disguise. Count the matchsticks for the first three patterns and you have a number sequence.

Why children guess

They look at two numbers, not four. A child sees 2 and 4, decides the rule is "double it", and writes 8. The sequence was 2, 4, 6, 8, going up by 2. Checking the rule against every number given takes five seconds and saves the mark.

They count along to a far term. Asked for the 50th term, a child adds 3 over and over. It works, slowly, until one addition goes wrong and every number after it is wrong too. The nth term rule gets there in one step.

They write the jump as the rule. A sequence that goes up by 3 gets the rule n + 3. The jump tells you to multiply the position by 3, so the rule starts with 3n.

They count the numbers instead of the steps. In 7, ?, ?, 19 there are four numbers but only three steps between them. Dividing the gap of 12 by 2, or by 4, gives the wrong answer.

The method, step by step

Take the sequence 5, 8, 11, 14.

1. Write the difference above each gap. 8 minus 5 is 3, and so is each gap after it. Your child writes +3 above every gap on the paper.
2. Check the difference is the same all the way along. If it is, the sequence is linear and the rest of this method works. If it is not, look at the other kinds of sequence further down.
3. Start the rule with the difference times n. The difference is 3, so write 3n. The letter n stands for the position in the sequence, 1st, 2nd, 3rd and so on.
4. Adjust to fit. The 3 times table goes 3, 6, 9, 12. The sequence is 5, 8, 11, 14, which is 2 more each time. So the rule is 3n + 2.
5. Check with a term you know. The 4th term should be 3 times 4, plus 2, which is 14. It is, so the rule is right.

Now the far away term is one sum. The 50th term is 3 times 50, plus 2, which is 152.

The same rule answers the "is this number in the sequence" question. For 50, solve 3n + 2 = 50. Take away 2 to get 3n = 48, then divide by 3 to get n = 16. That is a whole number, so 50 is the 16th term. For 60, 3n = 58, and 58 does not divide by 3 exactly, so 60 is not in the sequence.

Missing terms in the middle

For 7, ?, ?, 19, count the steps from 7 to 19. There are three, one into each empty box and one into the 19. The total change is 19 minus 7, which is 12, and 12 divided by 3 steps is 4. So the missing numbers are 11 and 15.

The number of steps is always one more than the number of empty boxes. Once your child says that out loud a few times, these questions stop being guesses.

Sequences that don't go up by the same amount

When the differences change, your child should look for one of these before anything else.

• Square numbers, 1, 4, 9, 16, 25, where the differences go 3, 5, 7, 9.
• Triangular numbers, 1, 3, 6, 10, 15, where the differences go 2, 3, 4, 5.
• Multiplying sequences, such as 3, 6, 12, 24, where each term is double the last. These are called geometric sequences.
• Add the two before, such as 1, 1, 2, 3, 5, 8, where each term is the sum of the two before it.

Writing the differences above the gaps still helps here. A child who sees 3, 5, 7, 9 written out will usually spot the square numbers faster than one staring at 1, 4, 9, 16.

Practising at home

Sequences are good for five minute games because they need no equipment.

Start and jump. You give a start number and a jump, such as start at 4 and add 6, and your child says the next five terms. Then swap, and they give you one.

Rule detective. Write four terms and ask for the rule, the 10th term and the 100th term. The 100th term is the one that shows whether they are using the rule or counting.

Matchstick squares. Make a row of one square with four matchsticks or pencils, then two squares, then three. The counts go 4, 7, 10, so the rule is 3n + 1, and ten squares need 31. Building it on the table first makes the rule feel real.

Question 11 on our free algebra practice page is always an nth term question, with fresh numbers each time you press New questions. For step by step practice, the Maths Skills Lab has a linear nth term skill that walks through the method one press at a time.

The lesson we built for this

Our Sequences Mastery lesson has 7 units and over 100 questions, in this order: finding the next term, finding missing terms, geometric sequences, the nth term rule, the nth term rule with an adjustment, finding a term from the rule, and checking whether a number is in the sequence.

Each unit opens with a worked example your child clicks through one line at a time, then practice questions marked easy, medium and hard with the answers hidden until they click. A 20 question challenge at the end mixes all seven units.

It is one of the ten lessons in the Algebra Mastery Bundle, and lesson 47 of The Complete 11+ Maths Course.

What to do this week

Two questions will show you where your child is.

Ask for the 20th term of 4, 7, 10, 13. The answer is 61, because the rule is 3n + 1. If your child counts along, they may get there, so follow up by asking for the 100th term. The answer is 301. If they cannot do that without counting, the nth term is the next thing to teach.

Then ask them to fill the gaps in 6, ?, ?, ?, 26. The answer is 11, 16, 21, because 20 divided by 4 steps is 5. If they divide by 3, they are counting the empty boxes instead of the steps.

For the other parts of algebra, our guide to 11+ maths at home has a quick check question, and the 11+ algebra help post covers where children get stuck with letters and equations.

Aadam, SHLC Tutors