Patterns

Mathematics is the study of numbers and the many ways in which they are written. All numbers in mathematics have a certain relationship which is defined by different patterns - numerical patterns, picture patterns, logical patterns, word patterns, etc. The number pattern is the most basic and commonly used, which includes odd-even numbers, skip counting, and other concepts. Whereas, more complex patterns form the basis of advanced mathematics.

But what are patterns and why do we need to know about them? Well, a series or sequence that follows a certain order is called a pattern in mathematics. We need to know about patterns because they are all around us - in plants, nature, behaviour, and not just numbers. 

In simple words, number patterns are all predictions. Let’s discuss a few examples of numerical patterns :

1. Pattern of even numbers -: 2, 4, 6, 8, 10, 1, 14, 16, 18, 20…

2. Pattern of odd numbers -: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21….

3. A pattern of numbers increasing by 5: 5, 10, 15, 20, 25.….

We are going to discuss the definition of pattern in Mathematics with a few solved example problems. In this article, we are going to focus on various patterns and pattern definitions in maths.

Pattern Definition

Math patterns are basically sequences that repeat according to rules. In maths, a rule can be defined as a set way to calculate or solve a problem. In Mathematics, the patterns can be related to any type of event or object. If the set of numbers is related to each other in any specific kind of rule, then the rule or manner is known as a pattern. They can be finite or infinite in numbers.

For example, in the given sequence 2, 4, 6, 8,? Each number is increasing by 2. So, the last number will be 8 + 2 which is equal to 10.

Number Patterns

According to the definition of pattern, there is a common type of maths pattern known as the number pattern. Number patterns can be defined as a sequence of numbers that are ordered based upon a rule. A number pattern establishes the same relationship between different numbers. Now, according to the pattern definition, there are many ways to figure out the rule, such as:

• Use a number line to see the distance or the difference between the numbers or what the numbers have in common.

• Look at the last one or two digits or the first digit of the numbers to see if they repeat in any special manner.

• Look at the numbers and see if there is any pattern, like taking each number and multiplying by 2 for instance.

• Think about the common number patterns, like counting by 2s, 5s, or 10s.

• You can also find the difference between the numbers.

It's important to remember that a number pattern can have more than one solution and a combination of rules present. In this case, try to think of the simplest rule that is possible, like adding 2 or multiplying by 3 with a difference of 4.

Types of Number Patterns

The main types of number patterns are:

• The Arithmetic/ Algebraic Pattern

• The Geometric Pattern

While, there are also four special sequences of number patterns, which are:

• The Fibonacci Pattern

• The Triangular Number Pattern

• Square Number Pattern

• Cube Number Pattern

Types of Modes of Number Patterns

Apart from number patterns, the three types of patterns commonly used in discrete mathematics are:

1. Repeating Patterns – If the number pattern changes in the same value every time, then the pattern can be known as a repeating pattern. Example: 1, 2, 3, 4, 1, 2, 3, 4,...

2. Growing Patterns – If the numbers are present in an increasing form, then the pattern can be known as a growing pattern. Example 34, 40, 46, 52, 56…..

3. Shirking Patterns – In the shirking pattern, the numbers are generally in decreasing form. Example: 42, 40, 38, 36, 32..

Rules for Patterns in Maths

Now, to construct or even decode a pattern, we have to know about some of the rules. The set of rules can be applied only after the knowledge of the type of sequence and the difference between two numbers. To find the numerical patterns, remember these two important rule categories:

1. Numbers of a particular pattern are considered to be in increasing order as they become larger. Typically, these patterns involve addition or multiplication.

2. Numbers of a particular pattern are considered as descending sequences as they become smaller. Division or subtraction is often involved in these patterns.

By considering these in a numerical pattern, you would either have to find the next number in a sequence by finding the difference between successive terms or you can find the next missing term by cracking the pattern between different terms of the series. 

Solved Questions 

Example 1. What pattern rule is the given series following? 81, 27, 9.

Ans. We can see that it is a descending pattern and the numbers are a multiple of 9.

So, taking the smallest number 9. 

From the series,  9 × 3 = 27

Ans, 9 x 9 = 81

Or we can say, 27 x 3 = 81

Now, going by the descending order we can conclude that the series is showing the pattern rule of: "divide by 3".

Example 2. Find the value of A in the sequence: 1, 8, A, 64, 125

Ans. In the series, the observable pattern is that every number is the perfect cube of the consecutive counting numbers. 

13 = 1, 23 = 8, 43 = 64, 53 = 125

Hence, the missing number 'A' is a cube of 3 which is 27.

Example 3. Find the missing value in the pattern: 3, __, 48, 192, __

Ans. Considering the two consecutive terms 48 and 192, we can see that,

48 x 4 = 192

So, the pattern rule for the series is: ‘multiply by 4.’

First missing term = 3 x 4 = 12

Second missing term = 192 x 4 = 768