Can a pattern have more than one rule?

With only two or three terms, yes, several rules can fit. That is why you need enough terms and must check your rule on all of them. With $2, 4$ the rule could be $+2$ or $\times 2$, but $2, 4, 8$ fits only $\times 2$.

Why do I add 9 steps and not 10 to get the 10th term?

The first term is already in place. To go from term $1$ to term $10$ you make $9$ jumps, just like walking from the 1st to the 10th step of a ladder.

How do I find the 20th shape in a repeating pattern?

Count the shapes in one block. For circle, square, triangle the block has $3$ shapes. $20 \div 3 = 6$ remainder $2$, so the 20th shape is the 2nd shape of the block: a square.

Teacher Resources

Pattern Recognition

Learn to spot the rule behind number and shape patterns, check it against every term, and predict what comes next.

Elementary20 minLesson

Printable worksheet

Definition

A pattern is a list of numbers or shapes that follows a rule. Each item in the list is called a term.

To recognize a pattern, look at how each term changes into the next one:

Adding pattern: the same number is added each time. $5, 9, 13, 17$ (add $4$ )
Subtracting pattern: the same number is taken away each time. $30, 25, 20, 15$ (subtract $5$ )
Multiplying pattern: each term is multiplied by the same number. $3, 6, 12, 24$ (multiply by $2$ )
Alternating pattern: two steps take turns. $10, 13, 11, 14, 12$ (add $3$ , subtract $2$ )
Growing pattern: the step itself grows. $1, 2, 4, 7, 11$ (add $1$ , then $2$ , then $3$ , then $4$ )
Repeating pattern: a block repeats again and again, like circle, square, triangle, circle, square, triangle

A rule only counts if it works for every term, not just the first two. Finding a rule from examples is inductive reasoning; checking it on all terms keeps you honest.

Try it now

What is the next number in the pattern

2, 5, 8, 11, \ldots

Worked Examples

Find the rule and the next two terms: $5, 9, 13, 17, \ldots$

Find the change between neighbors

$9 - 5 = 4$ , $13 - 9 = 4$ , $17 - 13 = 4$ → The difference is always $4$

State the rule

Every term is $4$ more than the one before → Rule: start at $5$ , add $4$

Use the rule to continue

$17 + 4 = 21$ , $21 + 4 = 25$ → $21$ and $25$

Answer: The rule is add $4$ . The next two terms are $21$ and $25$ .

Common Mistakes

Choosing a rule after looking only at the first two terms

Why it's wrong: In $2, 4, 8, 16$ the first jump is $+2$ , but $4 + 2 = 6$ , not $8$ . Many rules fit two numbers.

Correct: Test the rule on every pair of neighbors. Here $\times 2$ works for all of them: $2, 4, 8, 16$ .

Using $10$ jumps to reach the 10th term

Why it's wrong: The first term is already there. From term $1$ to term $10$ there are only $9$ jumps.

Correct: 10th term $=$ first term $+ 9 \times$ step. For $4, 7, 10, \ldots$ : $4 + 9 \times 3 = 31$ , not $4 + 10 \times 3 = 34$ .

Thinking every pattern adds the same number

Why it's wrong: In $2, 6, 18, 54$ the differences are $4, 12, 36$ , so it is not an adding pattern.

Correct: If the differences are not equal, try multiplying ( $\times 3$ here), alternating steps, or growing differences.

Forgetting where you are in an alternating pattern

Why it's wrong: In $10, 13, 11, 14, 12, 15$ the next step depends on the last one. Using $+3$ again gives $18$ , which is wrong.

Correct: Write the changes under the terms ( $+3, -2, +3, \ldots$ ) and continue the turn: $15 - 2 = 13$ .

Interactive Visual

Sequence Explorer

First Term (a₁ = 4)

Common Difference (d = 3)

10,

13,

16,

19,

22,

25,

28,

, ...

a_1

a_5

a_10

a_20

Sequence Explorer

11,

16,

, ...

Interactive Sandbox

Expression Calculator

Try these:

History

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Practice Problems

18 problems

Problem 1 of 18

Easy

What is the next number in the pattern $2, 5, 8, 11, \ldots$ ?

Why It Matters

Patterns are everywhere, and spotting them saves time:

Calendars: every Monday of a month is $7$ days after the last one, so you can find any date without counting day by day
Tiles and building: if each new row of a floor needs $3$ more tiles, you can plan how many tiles to buy
Saving money: putting the same amount in a piggy bank each week lets you predict when you reach your goal
Music and art: rhythms and borders are repeating patterns
Science and computers: scientists look for patterns in data, and computers are trained to recognize patterns in pictures and sounds

Pattern recognition is the first step of logical thinking: you notice a rule, test it, and then use it to predict.

Real World Applications

Calendars

The same weekday comes back every $7$ days, so dates of that weekday form an adding pattern.

Example:

If the first Monday of a month is the $3$ rd, the Mondays are the $3$ rd, $10$ th, $17$ th, $24$ th and $31$ st.

1Try It Yourself

The first Saturday of a month is the $6$ th. Swimming lessons are on every Saturday.

What date is the 4th Saturday of the month?

Step 1: Write the mathematical expression

Start at $6$ and add $3$ jumps of $7$ days:

Tiles and Patterns in Building

Builders and designers use tile patterns for floors, paths and walls, and they need to know how many tiles each part uses.

Example:

A path where each new section adds $2$ tiles uses $6, 8, 10, 12, \ldots$ tiles. The builder can predict any section without drawing it.

2Try It Yourself

A tile design has $5$ tiles in Figure 1, $8$ tiles in Figure 2 and $11$ tiles in Figure 3.

How many tiles are in Figure 6?

Step 1: Write the mathematical expression

Start at $5$ and add $5$ jumps of $3$ :

Saving Money

Saving the same amount every week makes the total in a piggy bank grow in an adding pattern.

Example:

With 10 dollars at the start and 4 dollars saved each week, the totals are 14, 18, 22, 26 dollars after weeks 1 to 4.

3Try It Yourself

Nora has 15 dollars in her piggy bank. She puts in 5 dollars every week.

How much money will she have after 8 weeks?

Step 1: Write the mathematical expression

Start at $15$ and add $8$ jumps of $5$ :

Key Takeaways

1A pattern follows a rule, and each item in it is called a term
2First look at the differences between neighbors: equal differences mean an adding or subtracting pattern
3If the differences are not equal, try multiplying, alternating steps or growing differences
4A rule is only correct if it works for every term, not just the first two
5To get the 10th term of an adding pattern, add $9$ steps to the first term
6In a repeating pattern, divide the position by the length of the block and use the remainder

Frequently Asked Questions

Write the difference between each pair of neighboring terms under the list. If they are all the same, you have an adding or subtracting pattern. If not, try multiplying or look for a pattern in the differences.

Glossary

Pattern: A list of numbers or shapes that follows a rule
Term: One item in a pattern; the 1st term is the first item, the 2nd term the next one, and so on
Rule: The instruction that tells you how to get from one term to the next, such as add $4$ or multiply by $2$
Difference: The result of subtracting one term from the next one, for example $9 - 5 = 4$
Alternating pattern: A pattern in which two different steps take turns, such as $+3$ , $-2$ , $+3$ , $-2$
Repeating pattern: A pattern in which the same block of items repeats, such as red, blue, red, blue

Formula Card

Difference check

\text{difference} = \text{next term} - \text{term}

If all differences are equal, the pattern adds (or subtracts) that number

10th term of an adding pattern

\text{10th term} = \text{1st term} + 9 \times \text{step}

From the 1st to the 10th term there are 9 jumps

Position in a repeating pattern

\text{position} \div \text{block length} = \ldots \text{ remainder } r

The remainder $r$ tells you which item of the block it is; remainder $0$ means the last item

Pattern Recognition

Definition

Worked Examples

Common Mistakes

Interactive Visual

Sequence Explorer

Sequence Explorer

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Calendars

Tiles and Patterns in Building

Saving Money

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Difference check

10th term of an adding pattern

Position in a repeating pattern

More in This Topic

Introduction to Logical Reasoning

Deductive Reasoning

If-Then Statements

Converse and Inverse