Number Patterns: Definition, Types, Facts and Examples

It tells the pattern or sequence in the given series of numbers. The number pattern tells the common relationship between the given set of numbers. A repeating arrangement of numbers with a certain rule is known as a number pattern.

Knowledge of numbers is the basic requirement to learn Mathematics. There are different types of numbers, such as natural, even, and odd numbers, prime and composite numbers, fractional numbers, decimal numbers, integers, rational and irrational numbers, etc.

Numbers and Primary Patterns

The number is a basic concept in Mathematics. In our daily life, we use numbers for counting. All counting numbers are known as natural numbers. The natural numbers, along with the zero, is called whole numbers. Thus, natural numbers start from one, and whole numbers start from zero.

Even and Odd Numbers

One of the basic patterns of the numbers is even and odd numbers. Numbers, which are exactly divisible by two \(\left( 2 \right),\) are known as even numbers. Even numbers leave the remainder zero when it is divided by two.
Even numbers are generally in the form of \(2\,n,\) where \(n = 1,\,2,\,3,\,4,\,…\)
Example: \(2,\,4,\,6,\,8,\,10,\,12,\,14,\,16,\,…\)
The numbers, which are not divisible by two \(\left( 2 \right),\) are known as odd numbers. Odd numbers leave the remainder one \(\left( 1 \right)\) when it is divided by two.
Odd numbers are generally in the form of \(2\,n + 1,\) where \(n = 0,\,1,\,2,\,3,\,….\)
Example: \(1,\,3,\,5,\,7,\,9,\,11,\,13,\,15,\,…\

Prime and Composite Numbers

One of the most used number patterns in our primary classes is prime and composite numbers. The numbers which have only two factors, such as one and themselves, are called prime numbers.
Example: \(2,\,3,\,5,\,7,\,11,\,13,\,17,\,….\)
\(2\) is the only even prime number and also the least prime number.
The numbers which have more than two factors are called composite numbers. \(4\) is the least composite number.
Example: \(4,\,6,\,8,\,9,\,10,\,12,\,…\)

In the above table, the blue colour marked tells the prime numbers, the white colour marked tells the composite number, and the red colour marked number \(\left( 1 \right)\) is neither prime nor composite.

Definition of Number Patterns

Number Pattern is the pattern or sequence in the given series of numbers. The number pattern tells the common relationship between the given set of numbers.  A repeating arrangement of numbers with a certain rule is known as a number pattern.

In our daily life, we observe the patterns of colours, shapes, actions and numbers, etc. In mathematics, a set of numbers are arranged in a series, and they are related to each other with a specific rule.

Example:
In the series \(1,\,3,\,5,\,7,\,9\) the next number is \(11.\) Here, we can understand that a given pattern of numbers is odd numbers, and each number is obtained by adding \(2\) to the previous number.

Order of Number Patterns

In the number patterns, the generally used concept is ordering. In our primary classes, we studied ordering the objects, numbers according to their values.

There are two types of ordering: They are ascending order and descending order.

Ascending Order

The series of numbers, which are written in the order of smallest number to the largest number is known as ascending order. In ascending order, the smallest number comes first and the largest number last.
Example: \(1,\,2,\,3,\,4,\,5,\,6,\,….\)

Descending Order

The series of numbers, which are written in the order of the largest number to the smallest number is known as descending order. In descending order, the largest number comes first, and the smallest number comes last.
Example: \(100,\,99,\,98,\,97,\,96,\,……\)

Types of Number Patterns

In mathematics, number patterns tell the set of arranged numbers in a series, and they are related to each other with a specific rule.

There are different number patterns, including arithmetic and algebraic patterns, geometrical pattern, and the Fibonacci number pattern.

The following figure tells the number of patterns that we commonly used.

Arithmetic Number Pattern

An arithmetic pattern or sequence is one type of arithmetic pattern. An arithmetic pattern is a series of numbers related to each other by a constant addition or subtraction.

An arithmetic pattern is a sequence where each term is obtained by adding a constant number to the previous term (Except the first term).
Here, the constant number is called a “common difference”, and it is represented by \(d.\)

Let the first term of A.P be \({a_1},\) then
The second term is \({a_1} + d.\)
And, the third term is \({a_1} + d + d = {a_1} + 2\,d\)
The fourth term is \({a_1} + 2\,d + d = {a_1} + 3\,d\) and so on.

Example:
For series \(6,\,13,\,20,\,27,\,33,\,….\)
The first term is \(6.\)
Here, each term is obtained by adding \(7\) to the previous term.
Thus, the common difference is \(7.\)

Geometric Pattern

A geometric pattern or sequence is one type of algebraic pattern. The geometric pattern is a series of numbers related to each other by a constant multiplication or division.

A geometric progression is a sequence where each term is obtained by multiplying a constant number to the previous term (Except the first term).

Here, the constant number is called as “common ratio”, and it is represented by \(r.\)

Let the first term of G.P is \(a,\) then
The second term is \(ar.\)
And, the third term is \(ar \times r = a{r^2}\)
The fourth term is \(a{r^2} \times r = a{r^3}\) and so on.

Example:
For the series \(1,\,2,\,4,\,8,\,16,\,32,\,…\)
The first term is \(1.\)
Each term is obtained by multiplying \(2\) to the previous term.
Thus, the common ratio is \(2.\)

Harmonic Pattern

Harmonic pattern or sequence is the sequence of numbers in which all the numbers are reciprocal of the numbers of an arithmetic progression.

Example:
The reciprocals of natural numbers \(\left( {1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,…} \right)\) gives the harmonic sequence.
\(1,\,\frac{1}{2},\,\frac{1}{3},\,\frac{1}{4},\,\frac{1}{5},\,\frac{1}{6},\,\frac{1}{7},\,\frac{1}{8},\,….\)

Triangular Number Pattern

A triangular number pattern is the sequence of numbers used to form some patterns in the form of an equilateral triangle. Dots represent the numbers in the triangle number patterns.

The numbers \(1,\,3,\,6,\,10\) and so on are the triangular numbers, which form an equilateral triangle pattern.

Square Number Pattern

A square number is a number obtained by multiplying a number by itself. The square number pattern represents the sequence of numbers, which are perfect squares. A square number pattern is a sequence in which the numbers are forming the pattern in the form of a square.

The numbers \(1,\,4,\,9,\,16,\,….\) and so on, are the square number pattern, such that \({1^1} = 1,\,{2^2} = 4,\,{3^2} = 9\) and \({4^2} = 16.\)

Cube Number Pattern

A cube number is a number obtained by multiplying a number by itself three times or a number multiplied by its square number. The cube number pattern is the sequence of numbers, which form the pattern in the form of the cube. A cube is a three-dimensional figure.

The numbers \(1,\,8,\,27,\,64,\,….\) and so on, are the cube numbers, such that \({1^3} = 1,\,{2^3} = 8,\,{3^3} = 27,\,{4^3} = 64.\)

Fibonacci Number Pattern

The Fibonacci pattern is one of the best formulas in mathematics. The Fibonacci pattern is the sequence of numbers in which each number is obtained by adding two terms preceding it. In the Fibonacci pattern, each number is the sum of its two previous numbers.

So, the sequence of numbers \(0,\,1,\,1,\,2,\,3,\,5,\,8,\,13,\,21,\,34,\,….\) etc. is called the Fibonacci sequence.

In the above figure, the red colour line drawn gives the Fibonacci number, the sum of the numbers on which it passes.

Modes of Number Patterns

In general, there are three types of modes of number patterns that are used in Mathematics. They are repeating, growing and shrinking pattern.

Repeating Pattern

A repeating pattern is the sequence of numbers in which numbers or patterns repeat over and over. In this pattern below, some numbers or a group of numbers are repeating.
Example:
\(1,\,2,\,3,\,4,\,5,\,1,\,2,\,3,\,4,\,5,\,1,\,2,\,3,\,4,\,5,\,….\)

Growing Pattern

A growing pattern is the sequence of numbers in which numbers are arranged in increasing order. In this sequence, numbers are arranged in ascending order.
Example:
\(1,\,3,\,5,\,7,\,9,\,….\)

Shrinking Pattern

A shrinking pattern is the sequence of numbers in which numbers are arranged in decreasing order. In this sequence, numbers are arranged in descending order.
Example:
\(25,\,23,\,21,\,19,\,…\)

Solved Examples – Number Patterns

Q.1. Find the missing value in the pattern given below:
\(1,\,4,\,9,\,16,\,\_\_,\,36,\,49,\,……\)
Ans:
Given pattern is \(1,\,4,\,9,\,16,\,\_\_,\,36,\,49,\,……\)
The given pattern is known as the pattern of square numbers.
From the above patterns, we can write \({1^2},\,{2^2},\,{3^2},\,{4^2},\,{5^2},\,{6^2},\,{7^2},\,…\)
So, the missing number in the pattern is \({5^2} = 25.\)

Q.2. What is the next number in the given arithmetic sequence \( – 12,\, – 7,\, – 2,\,3,\,….\)
Ans:
Given arithmetic sequence is \( – 12,\, – 7,\, – 2,\,3,\,….\)
We know that an arithmetic pattern is a sequence where each term is obtained by adding a constant number to the previous term (Except the first term).
Here, the constant number is called a “common difference”, and it is represented by \(d.\)
\(d = {a_2} – {a_1} = – 7 – \left( { – 12} \right) = – 7 + 12 = 5\)
The next number after \(3\) is obtained by adding a common difference \(\left( {d = 5} \right).\)
So, the next number is \(3 + 5 = 8.\)

Q.3. Find the value of \(x\) in the given series \(1,\,2,\,4,\,8,\,x,\,32,\,64,\,…..\)
Ans:
Given series is \(1,\,2,\,4,\,8,\,x,\,32,\,64,\,…..\)
The given series can be written as \(1,\,1 \times 2,\,2 \times 2,\,2 \times 2 \times 2,\,x,\,2 \times 2 \times 2 \times 2 \times 2,\,2 \times 2 \times 2 \times 2 \times 2 \times 2,\,…\)
\( = 1,\,2,\,{2^2},\,{2^3},\,{2^4},\,{2^5},\,{2^6},\,…\)
So, the value of \(x = {2^4} = 16.\)

Q.4. Calculate the value of \(P\) in the given series \(11,\,17,\,23,\,29,\,P,\,47,\,53.\)
Ans:
Given: \(11,\,17,\,23,\,29,\,P,\,47,\,53.\)
Here, the first number is \(11.\) Next, number \(17\) is obtained by adding 6 to the previous number \(11,\) such that\(17 = 11 + 6.\)
Similarly, the next number, \(23,\) is obtained by adding \(6\) to the \(17,\) such that \(23 = 17 + 6.\)
In the same way, the next number, \(P,\) is obtained by adding \(6\) to the \(29.\)
So, \(P = 29 + 6 = 35.\)

Q.5. Observe the next number in the given series \(1,\,3,\,6,\,….\)
Ans:
Given series is \(1,\,3,\,6,\,….\)
We know that the given sequence of numbers is triangular numbers.
In which each number is obtained by forming the numbers in the form of an equilateral triangle.

The triangular number series is \(1,\,1 + 2,\,1 + 2 + 3,\,1 + 2 + 3 + 4,\,….\)
So, the next number in the given series is \(1 + 2 + 3 + 4 = 10.\)

Summary

In this article, we have studied the number pattern, which tells the common relationship between the given set of numbers.  A repeating arrangement of numbers with a certain rule known as a number pattern is also discussed with examples. We have studied different modes of the pattern (ascending or descending).

We discussed different types of patterns, such as arithmetic patterns, geometric patterns, harmonic patterns, triangular patterns, square patterns, cube number patterns, Fabinocci series, discussed with examples.

Frequently Asked Questions(FAQ) – Number Patterns:

Q.1. What are the four types of sequence?
Ans: There are four types of sequences that are:
1. Arithmetic sequence
2. Geometric sequence
3. Harmonic sequence
4. Fibonacci sequence

Q.2. How to find number patterns?
Ans: A repeating arrangement of numbers with a certain rule is known as a number pattern.
To find the number patterns, we need to find the rule of the given numbers. Based on the rule, we will find the missing number or next number in the given series.

Q.3. What is the formula for geometric progression?
Ans: A geometric progression is a sequence where each term is obtained by multiplying a constant number to the previous term (except the first term).
Here, the constant number is called as “common ratio”, and it is represented by \(r.\)
The formula for geometric progression is \(a,\,ar,\,a{r^2},\,a{r^3},\,a{r^4},\,….,\,a{r^{n – 1}}\)

Q.4. What is the formula of arithmetic sequence:
Ans: An arithmetic pattern is a sequence where each term is obtained by adding a constant number to the previous term (except the first term).
Here, the constant number is called a “common difference”, and it is represented by \(d.\)
The formula of an arithmetic sequence is \(a,\,a + d,\,a + 2\,d,\,a + 3\,d,\,…. + a + \left( {n – 1} \right)d\)

Q.5. What are the rules of number patterns?
Ans: The rules of number patterns are:
1. The numbers, which are in increasing order are known as ascending order. In this type of sequence, the numbers generally involve addition or multiplication.
2. The numbers, which are in decreasing order are known as descending order. In this type of sequence, the numbers generally involve subtraction or division.

You can also refer to the NCERT Solutions for Maths provided by academic experts at Embibe for your final or board exam preparation.

We hope this detailed article on the Number Patterns has been informative to you. If you have any queries on this article, ping us through the comment box below and we will get back to you as soon as possible.