Pattern Recognition

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

Elementary20 min18 problems
Practice Problems

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. 5,9,13,175, 9, 13, 17 (add 44)
  • Subtracting pattern: the same number is taken away each time. 30,25,20,1530, 25, 20, 15 (subtract 55)
  • Multiplying pattern: each term is multiplied by the same number. 3,6,12,243, 6, 12, 24 (multiply by 22)
  • Alternating pattern: two steps take turns. 10,13,11,14,1210, 13, 11, 14, 12 (add 33, subtract 22)
  • Growing pattern: the step itself grows. 1,2,4,7,111, 2, 4, 7, 11 (add 11, then 22, then 33, then 44)
  • 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,…2, 5, 8, 11, \ldots?

Worked Examples

Find the rule and the next two terms: 5,9,13,17,…5, 9, 13, 17, \ldots
1

Find the change between neighbors

9−5=49 - 5 = 4, 13−9=413 - 9 = 4, 17−13=417 - 13 = 4 → The difference is always 44
2

State the rule

Every term is 44 more than the one before → Rule: start at 55, add 44
3

Use the rule to continue

17+4=2117 + 4 = 21, 21+4=2521 + 4 = 25 → 2121 and 2525

Common Mistakes

Choosing a rule after looking only at the first two terms

Why it's wrong: In 2,4,8,162, 4, 8, 16 the first jump is +2+2, but 4+2=64 + 2 = 6, not 88. Many rules fit two numbers.

Correct: Test the rule on every pair of neighbors. Here ×2\times 2 works for all of them: 2,4,8,162, 4, 8, 16.

Using 1010 jumps to reach the 10th term

Why it's wrong: The first term is already there. From term 11 to term 1010 there are only 99 jumps.

Correct: 10th term == first term +9×+ 9 \times step. For 4,7,10,…4, 7, 10, \ldots: 4+9×3=314 + 9 \times 3 = 31, not 4+10×3=344 + 10 \times 3 = 34.

Thinking every pattern adds the same number

Why it's wrong: In 2,6,18,542, 6, 18, 54 the differences are 4,12,364, 12, 36, so it is not an adding pattern.

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

Forgetting where you are in an alternating pattern

Why it's wrong: In 10,13,11,14,12,1510, 13, 11, 14, 12, 15 the next step depends on the last one. Using +3+3 again gives 1818, which is wrong.

Correct: Write the changes under the terms (+3,−2,+3,…+3, -2, +3, \ldots) and continue the turn: 15−2=1315 - 2 = 13.

Interactive Visual

Sequence Explorer

4,
7,
10,
13,
16,
19,
22,
25,
28,
31
, ...
12345678910
a_1

4

a_5

16

a_10

31

a_20

61

Sequence Explorer

1,
2,
4,
7,
11,
16,
22
, ...
1234567

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

18 problems
Problem 1 of 18
Easy
What is the next number in the pattern 2,5,8,11,…2, 5, 8, 11, \ldots?

Why It Matters

Patterns are everywhere, and spotting them saves time:
  • Calendars: every Monday of a month is 77 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 33 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 33rd, the Mondays are the 33rd, 1010th, 1717th, 2424th and 3131st.

1Try It Yourself

The first Saturday of a month is the 66th. Swimming lessons are on every Saturday.

What date is the 4th Saturday of the month?

Step 1: Write the mathematical expression

Start at 66 and add 33 jumps of 77 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 22 tiles uses 6,8,10,12,…6, 8, 10, 12, \ldots tiles. The builder can predict any section without drawing it.

2Try It Yourself

A tile design has 55 tiles in Figure 1, 88 tiles in Figure 2 and 1111 tiles in Figure 3.

How many tiles are in Figure 6?

Step 1: Write the mathematical expression

Start at 55 and add 55 jumps of 33:

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 1515 and add 88 jumps of 55:

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 99 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.
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.
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 2,42, 4 the rule could be +2+2 or ×2\times 2, but 2,4,82, 4, 8 fits only ×2\times 2.
The first term is already in place. To go from term 11 to term 1010 you make 99 jumps, just like walking from the 1st to the 10th step of a ladder.
Count the shapes in one block. For circle, square, triangle the block has 33 shapes. 20÷3=620 \div 3 = 6 remainder 22, so the 20th shape is the 2nd shape of the block: a square.

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 44 or multiply by 22
Difference
The result of subtracting one term from the next one, for example 9−5=49 - 5 = 4
Alternating pattern
A pattern in which two different steps take turns, such as +3+3, −2-2, +3+3, −2-2
Repeating pattern
A pattern in which the same block of items repeats, such as red, blue, red, blue

Formula Card

Difference check

difference=next term−term\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

10th term=1st term+9×step\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

position÷block length=… remainder r\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

Today's challengeOne problem, three tries

More in This Topic

Introduction to Logical Reasoning

Intermediate25 min

Deductive Reasoning

Intermediate25 min

If-Then Statements

Intermediate25 min

Converse and Inverse

Intermediate25 min