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# Patterns

Mathematics is mainly concerned about numbers.  The patterns play a very important role in a wide range of school mathematics. In math, we deal with different types of patterns, such as - number patterns, image patterns, logic, word pattern etc. Out of all, number patterns are very common in mathematics. These are quite familiar to the students who study maths frequently which later relates to functions. Several people  have spoken that math is the skill of patterns.

Few examples of mathematical numerical patterns are -
(1)  even numbers : 2, 4, 6, 8, 10, 1, 14, 16, 18, ...
(2)  odd number : 3, 5, 7, 9, 11, 13, 15, 17, 19, ...
(3)  Fibonacci numbers : 1, 1, 2, 3, 5, 8 ,13, 21, ...
(4) A difference of 5 : 0, 5, 10, 15, 20, 25, 30, 35, ...
and many more.

Let us talk about mathematical patterns in math with few solved problems which leads you to solve aptitude problems at your own pace in few seconds.

 Related Calculators Number Pattern Calculator

## What are Patterns in Math?

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In mathematics, the patterns are related to the type of any event or objects. If we contract a set of elements or numbers in which all these elements or numbers are related to each other in a specific rule, then this rule or manner is called the pattern. Sometimes, also known as sequence which are finite or infinite in numbers.

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

## Recursive Patterns in Math

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In recursive pattern rule, we get the next term by doing some addition, subtraction or applying some operation on the previous term. This patterns gives the information about the first(starting) number of the pattern and how the pattern continuous. In recursive pattern, we can easily get the next term with the help of previous one.

For example, if we have 5, 7, 9, 11, 13, 15,........, it is clear that this is the recursive pattern with the 5 at the starting number and 2. If we add 2 in the first term, then we get 7. If we add 2 in the second term(7), we get 9 and so on.

## Rules

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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 the pattern 4, 9, 16, 25, ?. In this pattern, it is clear that every number is the square of their position number and in the problem, we have n = 4 i.e. four term. The missing term take place at n = 5. So, if the missing is xn then xn = n2. Here, n = 5, then xn = (5)2 = 25.

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 and 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 in 1, we get 5 and if we add 4 in 5, we get 9 and so on. Sometimes, the first difference between the terms is not the same number, then we can check the second difference.

## Repeating Patterns Math

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The numbers which are present in the sequence change in the same way each time. These numbers can be subtracted or added. But, we can use the same amount or value each time for addition or subtraction.

Example 1: If we have a number pattern 0, 2, 4, 6, 8, ........ In this pattern, the staring term is 0 and add 2 each time to get the next number.

Example 2: If we have 32, 29, 26, 23, ........ This pattern starts with the value 32 and subtract 3 each time to get the new term.

## Types

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In discrete mathematics, we have three types:
1. Repeating
2. Growing
3. Shirking
If the numbers in a pattern change in the same way or same value each time, then that type of pattern is called repeating pattern. 1, 2, 3, 4, 5, ...... is the example for this. If the numbers are present in the increasing form, then the pattern is known as growing pattern. Example 34, 40, 46, 52, ..... is growing pattern as well as repeating also. In shirking pattern, the numbers are in decreasing form.

## Finding

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Given below are some of the examples that helps in understanding in better way:

### Solved Examples

Question 1: Learn the following number of sequence and calculate the value of P and the value of Q.

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

Solution:
85 79 73 67 61 55 49 43 P 31 25 Q

Here, number is decreasing by 6

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

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

Question 2: The given information value of the table below explain numbers positioned into groups I, II, III, IV, V and VI. In which grouping would be the following numbers belong?
1. 32
2. 33
3. 34
 I 1 7 13 19 25 II 2 8 14 20 26 III 3 9 15 21 27 IV 4 10 16 22 28 V 5 11 17 23 29 VI 6 12 18 24 30

Solution:
The remains when the number is divided by 6 determine the collection.
1. 26 + 6 = 32 remainder 3 (Group II)
2. 27 + 6 = 33 remainder 3 (Group III)
3. 28+ 6 = 34 remainder 5 (Group IV)

Question 3: A set of businessmen were at a networking meeting. Every businessman exchanges his business record with each additional businessman who was there. If there were 11 businessmen, how many business cards were exchanged?
Solution:
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55 exchanges

55 × 2 = 110 business cards.

If there were 11 businessmen, 110 business cards were exchanged.

## Number Patterns

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A list of numbers which follow a specific rule. This helps us to determine what number comes in next position in a given pattern. There are different types:
1. Arithmetic Pattern
2. Geometric Pattern
3. Fibonacci Pattern

### Solved Example

Question: Learn the following number of sequence and calculate the value of A and the value of B.

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

Solution:
15 22 29 36 43 A 57 64 71 78 85 B

Here, 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

## Geometric Patterns

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If we have a sequence 3, 9, 27, 81, .........., then we see that this a sequence with the factor of 3 between each number or we can say that we can get the next term by multiplying the previous one by number 3. This is called geometric sequence or pattern. Any pattern which is formed by multiplying particular number or value each time to the value of its previous one.

For example. 1, 2, 4, 8, ......, is the geometric pattern. We see that, 2 is the given multiplier and we get every successive term by multiplying 2 with it's previous term.
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