What is an arithmetic sequence?

An arithmetic sequence is a sequence in which the same number is added every time, so the gap between neighbouring terms never changes. That fixed gap is called the common difference. In 4, 10, 16, 22, the common difference is 6, and the next term is 28. A sequence that multiplies by the same number each time, such as 5, 10, 20, 40, is called geometric instead.

How do you find the missing number in a sequence?

You find the missing number by working out the rule from the terms you can see, then using it across the gap. Subtract neighbours to get the jumps; if they differ, divide neighbours to test for multiplying. In 3, 6, ?, 24, 48, every term doubles, so the gap holds 12. Then check the term after it: 12 × 2 = 24.

Is there a trick for spotting the rule in a number pattern?

The best trick is to study the jumps rather than the terms. Equal jumps mean the rule adds. Jumps that grow by 2 each time, like 3, 5, 7, 9 in 1, 4, 9, 16, 25, belong to the square numbers, each a number times itself. Jumps that grow by 1, as in 1, 3, 6, 10, 15, give the triangular numbers, which count dots stacked in a triangle.

Why does my child struggle with number patterns?

Many children find one jump and stop checking, which works for adding patterns but fails when the jumps change. In 1, 2, 4, 8, the first jump is 1, yet the rule is “double it”, so 16 comes next, not 9. Ask your child to test the rule on every pair before answering, and to say the rule aloud.

Is this number patterns game free, and how can a class use it?

Yes, every level, from counting in 2s to the formulas, is free in any browser, with nothing installed and no signup. For arithmetic sequence practice, write 7, 11, 15 on the board and let pupils add terms in turn until the row passes 50. Then ask for the 20th term without counting: the rule aₙ = 4n + 3 gives 83. The schools plan, priced per student, shows the teacher which pupils spot rules faster.

Math practice · reimagined

Number Sequences Game

This number sequence game climbs from counting by 2s, 5s and 10s to doubling and tripling patterns and formulas like aₙ = 6n + 1. Spot the rule at a glance and your fighter attacks; go term by term and your rival may hit first.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is a number sequence?

A number sequence is a list of numbers in a set order, where one rule fixes every number. Each number is called a term. The rule may add the same amount each time, multiply, or use the term’s position. Picture stepping stones at 3, 7, 11 and 15. Each jump is 4 long, so the next stone is 19.

Every level of Sequence

Example: fill the gap in 9, 14, ?, 24, 29

Find the jump between two neighbours you can see: 14 − 9 = 5, and 29 − 24 = 5 too.
So the row counts on in 5s, even though it starts at 9 rather than at 5 or 10.
Jump 5 on from the 14 just before the gap: 14 + 5 = 19.
Check against the number after the gap: 19 + 5 = 24, so you type 19.

Learn it step by step

A sequence is a line of numbers that follows a secret rule. Your job is to crack the rule and say what comes next.

In 2, 4, 6, 8, … each number jumps up by 2, so next is 10. Look at the gap between neighbours — if it's always the same, just add that gap again.

Type the next number in the pattern.

Tips and common mistakes

In a formula such as aₙ = 4n + 1, the number in front of n is the jump: its terms 5, 9, 13, 17 climb by 4 each time.
Measuring across the gap is a common mistake. In 4, 7, ?, 13, 16, the 6 from 7 to 13 covers two jumps of 3, so the answer is 10.
With buttons or dried beans, build piles of 2, then 5, then 8. Ask your child how many belong on the next pile, and how they know.

How to play

Five numbers line up with a question mark in third or fourth place, never last, as in 30, 40, ?, 60, 70, and you type 50 on the keypad. One level shows a formula instead: aₙ = 5n + 3, term 4 wants 23. Answer first and Mushroom strikes; a rule spotted early hits harder than one found late.

— Quick answers —

Sequence: the short FAQ

What is an arithmetic sequence?: An arithmetic sequence is a sequence in which the same number is added every time, so the gap between neighbouring terms never changes. That fixed gap is called the common difference. In 4, 10, 16, 22, the common difference is 6, and the next term is 28. A sequence that multiplies by the same number each time, such as 5, 10, 20, 40, is called geometric instead.
How do you find the missing number in a sequence?: You find the missing number by working out the rule from the terms you can see, then using it across the gap. Subtract neighbours to get the jumps; if they differ, divide neighbours to test for multiplying. In 3, 6, ?, 24, 48, every term doubles, so the gap holds 12. Then check the term after it: 12 × 2 = 24.
Is there a trick for spotting the rule in a number pattern?: The best trick is to study the jumps rather than the terms. Equal jumps mean the rule adds. Jumps that grow by 2 each time, like 3, 5, 7, 9 in 1, 4, 9, 16, 25, belong to the square numbers, each a number times itself. Jumps that grow by 1, as in 1, 3, 6, 10, 15, give the triangular numbers, which count dots stacked in a triangle.
Why does my child struggle with number patterns?: Many children find one jump and stop checking, which works for adding patterns but fails when the jumps change. In 1, 2, 4, 8, the first jump is 1, yet the rule is “double it”, so 16 comes next, not 9. Ask your child to test the rule on every pair before answering, and to say the rule aloud.
Is this number patterns game free, and how can a class use it?: Yes, every level, from counting in 2s to the formulas, is free in any browser, with nothing installed and no signup. For arithmetic sequence practice, write 7, 11, 15 on the board and let pupils add terms in turn until the row passes 50. Then ask for the 20th term without counting: the rule aₙ = 4n + 3 gives 83. The schools plan, priced per student, shows the teacher which pupils spot rules faster.

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