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Identifying Arithmetic Patterns of Numbers

Introduction

When we think of mathematics, the first thing that comes to our mind is numbers. We are aware of different kinds of numbers that have been defined such as natural numbers, whole numbers, decimal numbers, fractions and so on. Each set of a number has its own unique characteristic that makes it a set. For instance, the set of even numbers comprises of all numbers that are divisible by 2. Similarly, prime numbers are the numbers that are not completely divisible by any other number other than themselves and the number 1. By seeing these examples can we say that the numbers can be put in a sort of sequence or a pattern? Let us find out.

What is an Arithmetic Pattern?

An arithmetic Pattern is a sequence of numbers based on addition or subtraction to form a sequence of numbers that are related to each other. For example, if we have a sequence of numbers say 2, 4, 6, 8, and so on, we can clearly identify that this sequence is based on adding 2 to the previous number to obtain the new number. Hence, this sequence is a number pattern involving the addition of 2 to every digit.

Let us look at another example.

Suppose we have the sequence, 1, 4, 7, 10, 13. 16 ………

Can we identify the arithmetic pattern in this sequence?

Look at the sequence carefully. The following arithmetic pattern is observed –

First Number = 1

Second Number = 1 + 3 = 4

Third Number = 4 + 3 = 7

Fourth Number = 7 + 3 = 10

Fifth Number = 10 + 3 = 13

Therefore, we can say that the pattern in this sequence is “ 3 added to the previous number to obtain the next number ”.

Can we form a similar pattern by adding 5 to each number? Let us find out.

Suppose we have the series as shown in the pattern below –

We can see that starting from 0, there is a jump of every 5 numbers. This means that the above number pattern can be defined as the addition of 5 of the previous number to obtain the next number. The numbers in this pattern will be 0, 5, 10, 15, 20 and so on.

Types of Arithmetic Patterns

In the example above, we have seen that addition can be sued to generate an arithmetic pattern. Is this the only way to generate a number pattern? Following are the different types of arithmetic patterns that are most commonly in use –

• Growing Pattern – As the name suggests, the growing pattern is the arithmetic pattern where the numbers are present in an increasing order.
• Reducing Pattern – Again, as the name signifies, a reducing pattern is the arithmetic pattern in which the numbers are present in the decreasing order.
• Recurring Pattern – In the recurring pattern of numbers, the same set of numbers keep repeating themselves to form a pattern of numbers.

Now that we have understood the meaning of an arithmetic pattern, it is important to learn how to identify the same.

How to Identify Arithmetic Patterns of Numbers

We are aware of the four operations of mathematical operators, namely, addition, subtraction, multiplication and division.  Most of the arithmetic number patterns are based on these four mathematical operations only. However, there are some patterns that involve a combination of these operations.

Let us understand this by an example.

Suppose, we have been given the number pattern 1, 3, 5, 7, 9,  ………….

What arithmetic pattern is followed by the above sequence? Let us find out.

Observe each of the terms carefully. We can see that –

First Term = 1 = 2 x 0 + 1

Second Term = 3 = 2 x 1 + 1

Third Term = 5 = 2 x 2 + 1

Fourth Term = 7 = 2 x 3 + 1

Fifth Term = 9 = 2 x 4 + 1 and so on.

Therefore, we can identify the arithmetic pattern in the given sequence as 2 n + 1, where n ≥1. We can clearly see that this sequence involved a combination of two operators, “ x “ and “ + “.

Through this example, we have learnt that a number of different combinations of operators can be used to define the arithmetic pattern in a sequence. So, what do we do that helps us in the identification of the number pattern in a sequence? There are some standard patterns of numbers that we should be aware of. Let us discuss these common patterns.

Special Arithmetic Patterns of Numbers

There are some arithmetic patterns of numbers that are defined by the properties of the numbers included in them. Let us discuss about these special patterns –

Arithmetic Pattern of Odd Numbers

An arithmetic pattern of odd numbers is the sequence where we have all the numbers that are odd. Before we discuss the pattern, let us recall what we mean by odd numbers. Odd numbers are the numbers which when divided by 2 will leave 1 as a remainder. In other words, odd numbers are the numbers that are not divisible by 2. So, how is the pattern of odd numbers defined? The arithmetic pattern of odd numbers is given by 1 ,3 , 5 , 7 , 9  and so on. Mathematically, this pattern for natural numbers can also be represented as

Pattern of Odd Natural Numbers = 2 n + 1, where n 0.

Let us verify the above formula for obtaining the pattern of numbers.

If we put n = 0 in the above formula, we will have 2 x 0 + 1 = 0 + 1 = 1

If we put n = 1 in the above formula, we will have 2 x 1 + 1 = 2 + 1 = 3

If we put n = 2 in the above formula, we will have 2 x 2 + 1 = 4 + 1 = 5

If we put n = 3 in the above formula, we will have 2 x 3 + 1 = 6 + 1 = 7

If we put n = 4 in the above formula, we will have 2 x 4 + 1 = 8 + 1 = 9 and so on.

So, we can see that just by putting the value of the position of the number in the above formula, we can obtain the number in the pattern of odd numbers.

Arithmetic Pattern of Even Numbers

An arithmetic pattern of even numbers is the sequence where we have all the numbers that are even. Before we discuss the pattern, let us recall what we mean by even numbers. Even numbers are the numbers which when divided by 2 will leave 0 as a remainder. In other words, even numbers are the numbers that are completely divisible by 2. So, how is the pattern of even numbers defined? The arithmetic pattern of even numbers is given by 2, 4, 6, 8, 10 and so on. Mathematically, this pattern for natural numbers can also be represented as

Pattern of Even Natural Numbers = 2 n , where n 1.

Let us verify the above formula for obtaining the pattern of numbers.

If we put n = 1 in the above formula, we will have 2 x 1 = 2

If we put n = 2 in the above formula, we will have 2 x 2 = 4

If we put n = 3 in the above formula, we will have 2 x 3 = 6

If we put n = 4 in the above formula, we will have 2 x 4 = 8

If we put n = 5 in the above formula, we will have 2 x 5 = 10 and so on.

So, we can see that just by putting the value of the position of the number in the above formula, we can obtain the number in the pattern of even numbers.

Arithmetic Pattern of Triangular Numbers

An arithmetic pattern of triangular numbers is the pattern that has triangular numbers. But, what are these triangular numbers? Let us find out. Triangular numbers are generated from a pattern of dots that form a triangle. In other words, the triangular number sequence is the representation of the numbers in the form of an equilateral triangle. The pattern formed by the triangular numbers is such that the sum of the previous number and the order of the succeeding number results in the sequence of triangular numbers. This arrangement is represented as below –

So, the triangular pattern of numbers can be defined as 1, 3, 6, 10, 15, 21 and so on. Now, though we can easily identify the first few numbers of this pattern, how do we find a number on any position of the pattern? Can we define a formula to help us identify the number at a particular position in the number? The formula for defining the pattern of triangular numbers is given by

Arithmetic pattern of Triangular Numbers = $\frac{n ( n+1 )}{2}$ , where n≥1.

Let us verify this formula for the first few terms.

If we put n = 1 in the above formula, we will get $\frac{1 ( 1+1 )}{2} = \frac{2}{2} = 1$

If we put n = 2 in the above formula, we will get $\frac{ 2 ( 2+1 )}{2} = \frac{2 x 3}{2} = 3$

If we put n = 3 in the above formula, we will get $\frac{ 3 ( 3+1 )}{2} = \frac{3 x 4}{2} = 6$

If we put n = 4 in the above formula, we will get $\frac{ 4 ( 4+1 )}{2} = \frac{4 x 5}{2} = 10$

If we put n = 5 in the above formula, we will get $\frac{ 5 ( 5+1 )}{2} = \frac{5 x 6}{2} = 15$ and so on.

So, we can see that just by putting the value of the position of the number in the above formula, we can obtain the number in the triangular pattern.

Arithmetic Pattern of Square Numbers

An arithmetic pattern of triangular numbers is a pattern that has square numbers. But, what are these square numbers? Let us find out. Square numbers are the numbers obtained when a number is multiplied by itself. For instance 2 x 2 = 4, therefore, 4 is the square of 2. Similarly, 3 x 3 = 9, therefore, 9 is the square of 3. The arithmetic pattern of square numbers is given by 1, 4, 9, 16, 25, 36 and so on. Now, though we can easily identify the first few numbers of this pattern, how do we find a number on any position of the pattern? Can we define a formula to help us identify the number at a particular position in the number? The formula for defining the pattern of square numbers is given by

Arithmetic pattern of Square Numbers = n 2, where n≥1.

Let us verify the above formula for obtaining the pattern of numbers.

If we put n = 1 in the above formula, we will get 1 2 = 1 x 1 = 1

If we put n = 2 in the above formula, we will get 2 2 = 2 x 2 = 4

If we put n = 3 in the above formula, we will get 3 2 = 3 x 3 = 9

If we put n = 4 in the above formula, we will get 4 2 = 4 x 4 = 16

If we put n = 5 in the above formula, we will get 5 2 = 5 x 5 = 25 and so on.

So, we can see that just by putting the value of the position of the number in the above formula, we can obtain the number in the pattern of square numbers.

Arithmetic Pattern of Cube Numbers

An arithmetic pattern of triangular numbers is the pattern that has cube numbers. But, what are these cube numbers? Let us find out. Cube numbers are the numbers obtained when a number is multiplied twice with itself. For instance 2 x 2 x 2 = 8, therefore, 8 is the cube of 2. Similarly, 3 x 3 x 3 = 27, therefore, 27 is the cube of 3. The arithmetic pattern of cube numbers is given by 1, 8, 27, 12, 64, 125 and so on. Now, though we can easily identify the first few numbers of this pattern, how do we find a number on any position of the pattern? Can we define a formula to help us identify the number at a particular position in the number? The formula for defining the pattern of cube numbers is given by

Arithmetic pattern of Cube Numbers = n 3, where n≥1.

Let us verify the above formula for obtaining the pattern of numbers.

If we put n = 1 in the above formula, we will get 1 3 = 1 x 1 x 1 = 1

If we put n = 2 in the above formula, we will get 2 3 = 2 x 2 x 2 = 8

If we put n = 3 in the above formula, we will get 3 3 = 3 x 3 x 3 = 27

If we put n = 4 in the above formula, we will get 4 3 = 4 x 4 x 4 = 64

If we put n = 5 in the above formula, we will get 5 3 = 5 x 5 x 5 = 125 and so on.

So, we can see that just by putting the value of the position of the number in the above formula, we can obtain the number in the pattern of cube numbers.

Key Facts and Examples

1. An arithmetic Pattern is a sequence of numbers based on addition or subtraction to form a sequence of numbers that are related to each other.
2. A growing pattern is the arithmetic pattern where the numbers are present in an increasing order.
3. A reducing pattern is the arithmetic pattern in which the numbers are present in a decreasing order.
4. Recurring Pattern is the pattern of numbers, the same set of numbers keep repeating themselves to form a pattern of numbers.
5. A number of different combinations of operators can be used to define the arithmetic pattern in a sequence.
6. Odd numbers are the numbers which when divided by 2 will leave 1 as a remainder. Pattern of Odd Natural Numbers = 2 n + 1, where n≥1.
7. Even numbers are the numbers which when divided by 2 will leave 0 as a remainder. Pattern of Even Natural Numbers = 2 n , where n≥1.
8. Triangular numbers are generated from a pattern of dots that form a triangle. Arithmetic pattern of Triangular Numbers = $\frac{n ( n+1 )}{2}$ , where ≥1.
9. Square numbers are the numbers obtained when a number is multiplied with itself. Arithmetic pattern of Square Numbers = n 2, where n≥1.
10. Cube numbers are the numbers obtained when a number is multiplied twice with itself. Arithmetic pattern of Cube Numbers = n 3, where n≥1.