Patterns

Mathematics is all about numbers. It involves the study of different patterns. There are different types of patterns, such as number patterns, image patterns, logic patterns, word patterns etc. Number patterns are very common in Mathematics. These are quite familiar to the students who study Maths frequently. Especially, number patterns are everywhere in Mathematics. Number patterns are all predictions. In this article, we will discuss what is a pattern, and different types of patterns like, arithmetic pattern, geometric pattern and many solved examples. 

Table of Contents:

• Definition
• Types of Number Patterns
  • Arithmetic Pattern
  • Geometric Pattern
  • Fibonacci Pattern
• Rules
• Types
• Examples
• FAQs

Patterns in Maths

In Mathematics, a pattern is a repeated arrangement of numbers, shapes, colours and so on. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule, then the rule or manner is called a pattern. Sometimes, patterns are also known as a sequence. Patterns are finite or infinite in numbers.

For example, in a sequence 2,4,6,8,?. each number is increasing by sequence 2. So, the last number will be 8 + 2 = 10.

Few examples of numerical patterns are:

Even numbers pattern -: 2, 4, 6, 8, 10, 1, 14, 16, 18, …

Odd numbers pattern -: 3, 5, 7, 9, 11, 13, 15, 17, 19, …

Fibonacci numbers pattern -: 1, 1, 2, 3, 5, 8 ,13, 21, … and so on.

Number Patterns

A list of numbers that follow a certain sequence is known as patterns or number patterns. The different types of number patterns are algebraic or arithmetic pattern, geometric pattern, Fibonacci pattern and so on. Now, let us take a look at the three different patterns here.

Arithmetic Pattern

The arithmetic pattern is also known as the algebraic pattern. In an arithmetic pattern, the sequences are based on the addition or subtraction of the terms. If two or more terms in the sequence are given, we can use addition or subtraction to find the arithmetic pattern.

For example, 2, 4, 6, 8, 10, __, 14, __. Now, we need to find the missing term in the sequence.

Here, we can use the addition process to figure out the missing terms in the patterns.

In the pattern, the rule used is “Add 2 to the previous term to get the next term”.

In the example given above, take the second term (4). If we add “2” to the second term (4), we get the third term 6. 

Similarly, we can find the unknown terms in the sequence.

First missing term: The previous term is 10. Therefore, 10+2 = 12.

Second missing term: The previous term is 14. So, 14+2 = 16

Hence, the complete arithmetic pattern is 2, 4, 6, 8, 10, 12, 14, 16.

Geometric Pattern

The geometric pattern is defined as the sequence of numbers that are based on the multiplication and division operation. Similar to the arithmetic pattern, if two or more numbers in the sequence are provided, we can easily find the unknown terms in the pattern using multiplication and division operation. 

For example, 8, 16, 32, __, 128, __. 

It is a geometric pattern, as each term in the sequence can be obtained by multiplying 2 with the previous term.

For example, 32 is the third term in the sequence, which is obtained by multiplying 2 with the previous term 16.

Likewise, we can find the unknown terms in the geometric pattern.

First missing term: The previous term is 32. Multiply 32 by 2, we get 64.

Second missing term: The previous term is 128. Multiply 128 by 2, we get 256.

Hence, the complete geometric pattern is  8, 16, 32, 64, 128, 256.

Fibonacci Pattern

The Fibonacci Pattern is defined as the sequence of numbers, in which each term in the sequence is obtained by adding the two terms before it, starting with the numbers 0 and 1. The Fibonacci pattern is given as  0, 1, 1, 2, 3, 5, 8, 13, … and so on.

Explanation:

Third term = First term + Second term = 0+1 = 1

Fourth term = second term + Third term = 1+1 = 2

Fifth term = Third term + Fourth term = 1+2 = 3, and so on.

Rules for Patterns in Maths

To construct a pattern, we have to know about some rules. To know about the rule for any pattern, we have to understand the nature of the sequence and the difference between the two successive terms.

Finding Missing Term: Consider a pattern 1, 4, 9, 16, 25, ?. In this pattern, it is clear that every number is the square of their position number. The missing term takes place at n = 6. So, if the missing is x_n, then x_n = n². Here, n = 6, then x_n = (6)² = 36.

Difference Rule: Sometimes, it is easy to find the difference between two successive terms. For example, consider 1, 5, 9, 13,……. In this type of pattern, first, we have to find the difference between two pairs of the sequence. After that, find the remaining elements of the pattern. In the given problem, the difference between the terms is 4, i.e.if we add 4 and 1, we get 5, and if we add 4 and 5, we get 9 and so on.

Types of Patterns

In Discrete Mathematics, we have three types of patterns as follows:

• Repeating – A type of pattern, in which the rule keeps repeating over and over is called a repeating pattern.
• Growing – If the numbers are present in the increasing form, then the pattern is known as a growing pattern. Example 34, 40, 46, 52, …..
• Shirking – In the shirking pattern, the numbers are in decreasing form. Example: 42, 40, 38, 36 …..

Examples of Arithmetic and Geometric Pattern

Example 1:

Determine the value of P and Q in the following pattern.

85, 79, 73, 67, 61, 55, 49, 43, P, 31, 25, Q.

Solution:

Given sequence:85, 79, 73, 67, 61, 55, 49, 43, P, 31, 25, Q.

Here, the number is decreasing by 6

The previous number of P is 43. So, P will be 43 – 6, P = 37

The previous number of Q is 25. So, Q will be 25 – 6, Q = 19

Therefore, the value of P is 37 and Q is 19.

Example 2:

Determine the value of A and B in the following pattern.

15, 22, 29, 36, 43, A, 57, 64, 71, 78, 85, B.

Solution:

Given sequence: 15, 22, 29, 36, 43, A, 57, 64, 71, 78, 85, B.

Here, the number is increasing by +7

The previous number of A is 43. So, A will be 43 + 7, A = 50

The previous number of B is 85. So, B will be 85 + 7, B = 92

Therefore, the value of A is 50 and B is 92.

Example 3: 

Find the missing value in the geometric pattern: 120, 60, __, 15, __.

Solution:

Given: Geometric pattern is 120, 60, __, 15, __.

In this geometric pattern, the rule used is “Divide the previous term by 2 to get the next term”.

(i.e.,) 120/2 = 60

Then, the first missing term = 60/2 = 30

Second Missing term = 15/2 = 7.5

Hence, the geometric sequence is 120, 60, 30, 15, 7.5.

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