Number Sequences: How to Find the Rule

In Short: How Do You Solve Number-Sequence Problems?

• Find the rule first, then calculate. Write the next term only when you can explain how each term is produced.
• Check the differences between neighbouring numbers. A constant difference means you add or subtract the same amount each time.
• If the differences change, check multiplication, division, and then the differences between the differences.
• Check the rule against every given term, not just the first two.
• You can find a distant term without listing them all: add the common difference to the first term as many times as there are steps.

What Is a Number Sequence?

A number sequence is an ordered list of numbers that follows a rule. Each number is called a term: first term, second term, and so on. In 3, 7, 11, 15, the first term is 3 and the rule is “add 4”.

Children meet simple sequences in the early primary years when they count forwards and backwards in steps such as 2, 5, or 10. Later, mathematics curricula develop pattern recognition, generalisation, and algebraic rules. Number-sequence problems practise exactly those skills.

How Do You Find the Rule of a Sequence? Five Questions in Order

Ask these questions in order. Stop when one rule fits every term.

1. Is the difference constant? Write how much each term changes: 3 → 7 → 11 → 15 is +4 every time.
2. Is the ratio constant? If the numbers grow more quickly, check multiplication: 2, 6, 18, 54 is ×3 each time.
3. Do the differences form their own sequence? In 1, 2, 4, 7, 11, the differences are +1, +2, +3, +4. The next difference is +5.
4. Do two operations alternate? In 5, 8, 6, 9, 7, 10, the rule alternates between +3 and −2.
5. Are these familiar numbers? The sequence 1, 4, 9, 16, 25 contains square numbers: 1 × 1, 2 × 2, 3 × 3, and so on.

Write the differences in small numbers under the sequence. That often makes the pattern visible without holding every calculation in your head.

Number Sequences — Worked Examples

You find a missing term in the middle in the same way. In 4, 9, □, 19, 24, the common difference is 5, so the missing term is 14. In 50, 44, □, 32, 26, you subtract 6 each time, so the missing term is 38.

How Do You Find a Distant Term, Such as the 100th?

In a sequence with a constant difference, you do not need to write all 100 terms. Start with the first term and add the common difference once for each step. In 3, 7, 11, 15, there are 99 steps from the first term to the 100th, so the 100th term is 3 + 99 × 4 = 399.

Students around Grade 6–7 / Year 7–8 can express the same sequence algebraically. The nth term of 3, 7, 11, 15 is 4 × n − 1, where n is the term number. Check: n = 1 gives 3, n = 2 gives 7, and n = 100 gives 399.

Picture sequences work well too. Make a row of squares from matchsticks so adjacent squares share one side. One square uses 4 matchsticks, two use 7, and three use 10. Each new square adds 3 matchsticks, so 10 squares need 3 × 10 + 1 = 31 matchsticks. This moves children from counting examples to generalising a rule, which later supports algebra.

Problems to Solve Independently

Solve each one, then check the answers below. Write the rule in words as well as giving the next terms.

1. 2, 5, 8, 11, … — give the next two terms.
2. 81, 72, 63, … — give the next three terms.
3. 3, 6, 12, 24, … — give the next two terms.
4. 2, 4, 7, 11, … — give the next three terms.
5. 1, 10, 2, 20, 3, 30, … — give the next two terms.
6. How many matchsticks are needed for a row of 20 squares in the example above?
7. What is the 20th term of 5, 9, 13, 17, …?

Answers: 1) 14, 17 (+3); 2) 54, 45, 36 (−9); 3) 48, 96 (×2); 4) 16, 22, 29 (differences +2, +3, +4, +5, +6, +7); 5) 4, 40 (two interleaved sequences); 6) 3 × 20 + 1 = 61; 7) 5 + 19 × 4 = 81.

For younger children, begin with patterns made from pictures and colours. Activities such as continue the pattern for 5-year-olds prepare children for number sequences without calculations. Counting in 3s, 4s, or 7s also appears in multiplication facts to 100, so multiplication practice reinforces sequence thinking too.

What Mistakes Should You Avoid?

• Choosing a rule from only the first two terms. After 2, 4, the next term could be 6 under a +2 rule or 8 under a ×2 rule. More terms are needed to decide.
• Guessing without checking. The number you enter must fit a rule that can be stated clearly and applied to every term.
• Losing the sign. In a decreasing sequence, write negative differences: −9, −9, −9.
• Doing every calculation mentally. Writing differences under the sequence reduces avoidable mistakes.
• Assuming there is always one answer when only a few terms are given. After 1, 2, 4, both 8 under ×2 and 7 under differences +1, +2, +3 can be justified. When information is limited, write the rule beside your answer.

For more logical-thinking problems, see maths puzzles for kids, and for pattern ideas with blocks, see maths games with blocks.

How Can a Parent Practise Number Sequences With a Child?

A few minutes in everyday life is enough. Dates on the same weekday are 7 days apart, for example 3, 10, 17, 24. House numbers on one side of a street often increase by 2.

Swapping roles works well too. Your child invents a sequence with a hidden rule and you try to identify it. Creating a puzzle requires understanding the rule clearly.

Sources

• Podstawa programowa kształcenia ogólnego, matematyka, klasy IV–VIII — podstawaprogramowa.pl (aims: noticing regularities, writing relationships as algebraic expressions; version in force from the 2024/2025 school year).
• Podstawa programowa kształcenia ogólnego, edukacja wczesnoszkolna — podstawaprogramowa.pl (counting forwards and backwards from a given number in 1s, 2s and 10s).
• Khan Academy po polsku — Zadania o arytmetyce, 3. klasa (practice with rules, arithmetic patterns and sequence reasoning).
• Education Endowment Foundation — Improving Mathematics in Key Stages 2 and 3 (2017, updated 2022; guidance on examining the structure of patterns and generalising rather than relying only on repeated calculation).