What is Pattern in Math (Definition, Types & Examples)

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Mathematics is all about numbers. Is the pattern related to any concept in math? Yes, the pattern is an early building block in algebra – the study that helps represent problems or situations in the form of mathematical expressions. The word “pattern” means a series or sequence that generally repeats itself. You observe various patterns in your daily lives such as the ones of colours, actions, shapes, numbers, etc.

Let’s understand what is pattern and its different types.

What is Pattern in Math?

In math, a pattern is defined as a sequence of repeating objects, shapes, or numbers. We can relate a pattern to any type of event or object. A pattern has a rule that tells us which objects belong to the pattern and which objects do not belong to the pattern.

Let’s take a look at some patterns in math to understand their meaning.

Types of Patterns

The patterns can be categorized based either on the number of elements(or objects) or the type of elements(or objects) present in the pattern.

Based on the number of elements present, a pattern can be classified as 

• Finite Pattern
• Infinite Pattern

Based on the type of elements present, a pattern can be classified as 

• Shape Pattern
• Letter Pattern
• Number Pattern

What is a Finite Pattern?

A pattern is called a finite pattern when we know the first term(or element) and the last term(or element) of the pattern. In other words, a finite pattern contains a finite(or countable) number of elements.

For example, in the pattern $5$, $10$, $15$, $20$, $25$, $30$, the first term is $5$ and the last term is $30$. There are $6$ elements in the pattern.

What is an Infinite Pattern?

A pattern is called an infinite pattern when we know the first term(or element) but do not know the last term(or element) of the pattern. In other words, an infinite pattern contains an infinite(or uncountable) number of elements.

For example, in the pattern $5$, $10$, $15$, $20$, $25$, $30$, $35$, $40$, …, the first term is $5$ but we do not know where the pattern is going to stop. (The three dots (…) at the end of any pattern means it’s an infinite pattern).

What is a Shape Pattern?

A pattern is called a shape pattern when a group of shapes is repeated Shape patterns follow a certain sequence or order of shapes. The shapes can be simple shapes like squares, rectangles, triangles, circles, etc., or other objects such as flowers, arrows, moons, stars, etc.

Below are examples of a shape pattern.

In the above patterns, the arrow and the moon crescent are rotated at $90^{\circ}$ and change their colour. We can also say that each coloured shape is getting repeated after $2$ shapes.

What is a Letter Pattern?

A letter pattern is also called an alphabet pattern. A pattern is called a letter pattern that consists of English alphabets following a particular sequence. Or we can say that there exists a common relationship between any two adjacent letters(or alphabets).

Consider an example of A, C, E, G, I, K, M… In this pattern, one letter has been removed after every alphabet, or the letters are written starting from A and the next letters after skipping a letter.

Let’s consider one more example. A, E, I, M, Q, …. In this pattern, three letters have been removed after every alphabet, or the letters are written starting from A and the next letters after skipping a letter.

Or, in this pattern, Z, Y, X, W, V, U, …, the letters are written in reverse order starting from Z.

Examples

Ex 1: Find the next two terms of the given letter pattern $\text{A}$, $\text{CC}$, $\text{EEE}$, …

In this pattern, we notice two things 

First, the letters start from $A$, and the next letter is written by skipping one letter.

Second, the number of letters in each term increases by $1$.

Therefore, the next two terms are $GGGG$ and $IIIII$.

Ex 2: Find the next two terms of the given letter pattern $M$, $K$, $I$, $G$, …

In this pattern, we notice two things 

First, the letters are written in reverse order.

Second, the number of letters in each term is skipped by $1$.

Therefore, the next two terms are $E$ and $C$.

What is a Number Pattern?

A pattern is called a number pattern that consists of numbers following a particular sequence. Or we can say that there exists a common relationship between any two adjacent numbers. Number patterns are the most common patterns in mathematics.

The most common type of pattern in mathematics is the number pattern, where a list of numbers follows a certain sequence based on a rule. 

There are different types of number patterns:

Arithmetic Pattern

An arithmetic pattern is also known as an algebraic pattern. In this type of pattern, we start a pattern with a number and obtain the next number by adding or subtracting a fixed number to(or from) the previous term(or number).

The number we start with is called the first term represented by $a$ and a fixed number is called the common difference represented by $d$.

For example, $1$, $3$, $5$, $7$, $9$, … is an arithmetic pattern. Here the first term $a = 1$ and the common difference $d = 2$.

Notice that $3 – 1 = 5 – 3 = 7 – 5 = 9 – 7 = … = 2$.

Similarly, $22$, $18$, $14$, $10$, $6$, $2$, … is also an arithmetic pattern. Here the first term $a = 22$ and common difference $d = -4$. Here also a next term is obtained by adding $18 – 22 = 14 – 18 = 10 – 14 = 6 – 10 = 2 – 6 = … -4=$.

Notice that here we are adding $-4$ (or subtracting $4$) from each term to get the next term.

Examples

Ex 1: Find the next two terms of the given number pattern $2$, $\frac{5}{2}$, $3$, $\frac{7}{2}$, …

The first term of the number pattern is $2$.

The next number in the pattern is obtained by adding $\frac{1}{2}$ to the previous number.

$\frac{5}{2} = 2 + \frac{1}{2}$

$3 = \frac{5}{2} + \frac{1}{2}$

$\frac{7}{2} = 3 + \frac{1}{2}$

Therefore, the next two numbers are

$4 = \frac{7}{2} + \frac{1}{2}$

$\frac{9}{2} = 4 + \frac{1}{2}$

Ex 2: Find the next two terms of the given number pattern $46$, $39$, $32$, $25$, …

The first term of the number pattern is $46$.

The next number in the pattern is obtained by subtracting $7$ from the previous number.

$39 = 46 – 7$

$32 = 39 – 7$

$25 = 32 – 7$

Therefore, the next two numbers are

$18 = 25 – 7$

$11 = 18 – 7$

Geometric Pattern

In a geometric pattern, we start a pattern with a number and obtain the next number by multiplying or dividing a fixed number by the previous term(or number).

The number we start with is called the first term represented by $a$ and a fixed number is called the common ratio represented by $r$.

For example, $1$, $2$, $4$, $8$, $16$, … is a geometric pattern. Here the first term $a = 1$ and the common ratio $r = 2$.

Notice that $2 \div 1 = 4 \div 2 = 8 \div 2 = 16 \div 8 = … 2$.

Examples

Ex 1: Find the next two terms of the given number pattern $1$, $3$, $9$, $27$, …

The first term of the number pattern is $1$.

The next number in the pattern is obtained by multiplying $3$ by the previous number.

$3 = 1 \times 3$

$9 = 3 \times 3$

$27 = 9 \times 3$

Therefore, the next two numbers are

$81 = 27 \times 3$

$243 = 81 \times 3$

Ex 2: Find the next two terms of the given number pattern $62500$, $12500$, $2500$, …

The first term of the number pattern is $62500$.

The next number in the pattern is obtained by dividing $5$ by the previous number.

Therefore, the next two numbers are

$500 = 2500 \div 5$

$100 = 500 \div 5$

Harmonic Pattern

A harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. In other words, a pattern is called a harmonic pattern when the reciprocal of the terms form an arithmetic pattern. 

For example, $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{5}$, … is a harmonic pattern.

Notice that the reciprocal of the terms are $1$, $2$, $3$, $4$, $5$, … which is an arithmetic pattern where the next term is obtained by adding $1$ to the previous term.

Similarly, $1$, $\frac{1}{5}$, $\frac{1}{9}$, $\frac{1}{13}$, $\frac{1}{17}$, $\frac{1}{21}$, … is a harmonic pattern. The pattern formed by the reciprocal of the terms is $1$, $5$, $9$, $13$, $17$, $21$, … with a common difference $4$.

Examples

Ex 1: Find the next two terms of the given number pattern $1$, $\frac{1}{6}$, $\frac{1}{11}$, $\frac{1}{16}$, $\frac{1}{21}$, …

Taking the reciprocal of the terms of the pattern we get $1$, $6$, $11$, $16$, $21$, … which is an arithmetic pattern where the number is $1$ and the common difference is $5$. Therefore, the next two terms will be $21 + 5 = 26$ and $26 + 5 = 31$.

Thus, the next two terms of the pattern are $\frac {1}{26}$ and $\frac {1}{26}$.

Ex 2: Find the next two terms of the given number pattern $\frac {1}{7}$, $\frac {1}{10}$, $\frac {1}{13}$, $\frac {1}{20}$, …

Taking the reciprocal of the terms of the pattern we get $7$, $10$, $13$, $16$, … which is an arithmetic pattern where the number is $7$ and the common difference is $3$. Therefore, the next two terms will be $16 + 3 = 19$ and $19 + 3 = 22$.

Thus, the next two terms of the pattern are $\frac {1}{19}$ and $\frac {1}{22}$.

Fibonacci Pattern

A Fibonacci pattern is a sequence of numbers in which each number in the sequence is obtained by adding the two previous numbers together. This sequence starts with $0$ and $1$. We add the two numbers to get the third number in the sequence.   

The sequence $0$, $1$, $1$, $2$, $3$, $5$, $8$, $13$, … is the Fibonacci pattern. 

Notice that the pattern that is followed here is $0 + 1 = 1$,  $1 + 1 = 2$ , $1 + 2 = 3$ , $2 + 3 = 5$, $3 + 5 = 8$, $5 + 8 = 13$, ….

Read more about the Fibonacci pattern here.

Examples

Ex 1: Find the $7^{th}$ and $10^{th}$ terms of the Fibonacci series.

A Fibonacci pattern or series is obtained by starting with numbers $0$ and $1$ and adding the previous two numbers.

The first two terms are $0$ and $1$.

The $3^{rd}$ term = $0 + 1 = 1$

The $4^{th}$ term = $1 + 1 = 2$

The $5^{th}$ term = $1 + 2 = 3$

The $6^{th}$ term = $2 + 3 = 5$

The $7^{th}$ term = $3 + 5 = 8$

The $8^{th}$ term = $5 + 8 = 13$

The $9^{th}$ term = $8 + 13 = 21$

The $10^{th}$ term = $13 + 21 = 34$

Triangular Number Pattern

The representation of the numbers in the form of an equilateral triangle arranged in a series or sequence is known as a triangular number pattern. The numbers in the triangular pattern are in a sequence of $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$ and so on. 

The numbers in the triangular pattern are represented by dots. 

Examples

Ex 1: Find the $7^{th}$ term of the triangular series.

The triangular series is $1$, $3$, $6$, $10$, …

$1^{st}$ term = $1$

$2^{nd}$ term = $1 + 2 = 3$

$3^{rd}$ term = $1 + 2 + 3 = 6$

$4^{th}$ term = $1 + 2 + 3 + 4 = 10$

$5^{th}$ term = $1 + 2 + 3 + 4 + 5 = 15$

$6^{th}$ term = $1 + 2 + 3 + 4 + 5 + 6 = 21$

$7^{th}$ term = $1 + 2 + 3 + 4 + 5 + 6 + 7 = 28$

Rules of Pattern

As seen in the above example, for creating a complete pattern, a certain set of rules needs to be considered. For applying these rules, we should first understand the nature of the sequence and the difference between the two consecutive numbers given in the pattern. 

Based on the rule used in the pattern it can be categorized as either of the following three rules.

• Repeating Pattern: The type of pattern in which the rule keeps repeating over and over is called a repeating pattern. It is generally used in letter and shape patterns. For example, 
• Growing Pattern: The type of pattern in which the numbers are placed in increasing form or each number is greater than the previous number, then the pattern is known as a growing pattern. For example $12$, $17$, $22$, $27$, …
• Shrinking Pattern: The type of pattern in which the numbers are placed in the decreasing form or each number is smaller than the previous number. Example: $82$, $79$, $76$, $73$, $70$, …

Practice Problems

1. Write the next three terms of the following number patterns
   • $6$, $15$, $24$, $33$, $42$, …
   • $33$, $38$, $43$, $48$, $53$, …
   • $76$, $72$, $68$, $64$, $60$, …
   • $3$, $12$, $48$, $192$, …
   • $5$, $15$, $45$, $135$, …
2. Write the next two terms of the following letter patterns
   • $B$, $E$, $H$, $K$, …
   • $A$, $E$, $I$, $M$, …
   • $Z$, $X$, $V$, $T$, …

FAQs

Conclusion

The word “pattern” means a series or sequence that generally repeats itself. In this article, you learned what is pattern and how patterns with figures, letters, or numbers are formed using certain rules which generate either a growing, a shrinking, or a repeating pattern.

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)