Session 4: Patterns in Whole Numbers – Definition, Types and Examples

What do these images have in common?

They all have a pattern that repeats itself again and again in an identical way, sort of like a wallpaper pattern. Patterns exist all around us. Some patterns are decorative, while others may provide strength. Much of the mathematics we learn involves recognising and extending the pattern, analysing and describing a pattern or supplying some missing parts in the given pattern. One way to model patterns mathematically is with sequences. We will learn to recognise and tell patterns in whole numbers by finding a pattern rule in this concept.

Numerical Pattern

A numerical pattern is a sequence of numbers that have been created based on a formula or rule. Pattern rules can use one or more mathematical operations to describe the relationship between consecutive numbers in the pattern. We need to understand the nature of the sequence and the relation between the two consecutive numbers.
These are two primary categories of numerical patterns.

• When numbers in a pattern get larger as the sequence continues, they are in an ascending pattern. Ascending patterns often involve multiplication or addition. For example, in a sequence of4,8,12,16,…each number increases by 4. So, according to the pattern, the following number will be

 16+4=20.

• When the number in a pattern gets smaller, they are in a descending pattern as the sequence continues. Descending patterns often involve division or subtraction. For example, in a sequence of 48,42,36,30,…each number decreases by 6.
  So, according to the pattern, the following number will be

 30−6=24.

When given a pattern, we will often want to figure out the pattern rule that created the pattern.
1) Find the pattern rule for the sequence:

243,81,27,9,…

First, take an overview of the numbers. As the sequence continues, the numbers get smaller in value, which is a descending pattern. This means the rule likely involves division or subtraction.

Look at the smaller numbers at the end of the sequence.

Think: “What could we do to get 27 to get 9?”

We could subtract 18.

We could divide by 3.

Next, check if any of these potential pattern rules work with the rest of the sequence.

Consider 81 and 27.

If we subtract 18 from 81, we get 63, not 27. So the pattern rule is not “subtract 18”. If we divide 81 by 3, we get 27. So the pattern rule “divide by 3” seems to work. Now, make sure “divide by 3” works throughout the whole sequence.

“Divide by 3” works for the whole sequence.

So, the pattern rule is “divide by 3”.

Figuring out pattern rules can take some amount of guessing and checking. We will often have to come up with more than one potential pattern rule based on two of the numbers in the sequence and check which one works throughout the whole sequence.

Patterns in Whole Numbers

Observe the following patterns:

Numerical Pattern 1

Numerical Pattern 2

Numerical Pattern 3

Numerical Pattern 4

We can also express numbers based on geometrical shapes.
Depending on the geometrical shapes formed, different types of numbers are obtained. They are as follows:

1. Square numbers 
2. Triangular numbers
3. Cube numbers

Square Numbers

A square number is obtained by multiplying a number by itself. 1,4,9,16,25…… all are square numbers. The numbers themselves indicate their property.
For example,

It is read as 4 is equal to two squared or 4 is equal to two to the power two.
Similarly,

We call it 9 is equal to three squared.

Similarly,

Such numbers can be represented by dots arranged in rows and columns. For example, 4 can be represented by four dots in two rows and two columns, as shown below.

9 can be represented by nine dots in three rows and three columns.

Thus, the square numbers 1,4,9,… can be represented as follows:

The square numbers also follow a certain pattern of numbers.
For example,

1=1
4=1+2+1
9=1+2+3+2+1
16=1+2+3+4+3+2+1
25=1+2+3+4+5+4+3+2+1

The same numbers can be represented as the sum of odd numbers.
For example,

Triangular Numbers

The numbers that can be represented as triangles are called triangular numbers.
The triangular numbers follow a certain pattern of numbers.
For example,

1=1
3=1+2
6=1+2+3
10=1+2+3+4
15=1+2+3+4+5

Triangular numbers are made by arranging dots to form either equilateral or right-angled isosceles triangles.

Cube Numbers

The numbers obtained by multiplying the number thrice by themselves are called cube numbers.
Look at the following cube.

In geometry, the volume of a cube = length × width × height.
For the cube having a length of each side as a, we have

This means the volume of a cube is a cube number.
The cube shown above has a side of 1 unit.

Hence, a cube number is a product of multiplying a whole number by itself and then by itself again. Now, consider the following examples,