Informative line

Patterns And Their Rules

Pattern

  • Patterns exist everywhere around us.
  • A pattern is something that repeats in a specific way.
  • Definition: A pattern shows a relation between numbers or objects in which consecutive members are related to each other by a specific rule.
  • The rule tells us how the pattern repeats.

Examples:

(1) 1, 3, 5, 7, 9,........

In this example, there is a relation between every next term with the previous term. Every next term is increased by 2.

(2)  Consider the given figure.

In this example, there is a sequence of two circles after every triangle.

  • Patterns can be described in two ways:

(i) Number patterns

(ii) Figure patterns

Number Pattern:

A list of numbers that follows a certain sequence using a specific rule is called number pattern.

Ex. \(1,3,5,7,..... \\ 2,4,6,8,.....\\ 2,4,8,16,32,.....\)

Figure Pattern:

In this type of pattern, there can be two changing features:

(1) Number of shapes or objects

(2) Types of shapes

Illustration Questions

Which one is a number pattern in the following sequences?

A

B

C

D

×

We can observe that all the given sequences follow some specific rules.

Options (A), (C) and (D) are figure patterns because the feature that is changing in these is shapes and their orientation.

If numbers are changing by a specific rule, the pattern is a number pattern. 

In option (B), the numbers are changing according to a particular rule.

Hence, option (B) is correct.

Which one is a number pattern in the following sequences?

A image
B image
C image
D image

Option B is Correct

Number Pattern

A list of numbers that follows a certain sequence using a specific rule is called number pattern.

There can be different types of sequences.

For example:

1, 3, 5, 7, 9,....... Odd number series

2, 4, 6, 8, 10,.....Even number series

Multiples Series:

Series of 2's: 2, 4, 6, 8,.......

Series of 3's: 3, 6, 9, 12,.......

Series of 4's: 4, 8, 12, 16,.......

Series of 5's: 5, 10, 15, 20,.......

and so on.

Composite Number Series:

4, 6, 8, 9, 10, 12, 14, 15, 16,......

Prime Number Series:

2, 3, 5, 7, 11, 13, 17, 19, 23,......

Square Numbers:

1, 4, 9, 16, 25, ......

\(\Rightarrow\) \(1^2,\;2^2,\;3^2,\;4^2,\;5^2,...\) (Squares of the counting numbers)

Cube Numbers:

1, 8, 27, 64, 125, ......

\(\Rightarrow\) \(1^3,\;2^3,\;3^3,\;4^3,\;5^3,...\) (Cubes of the counting numbers)

Illustration Questions

Which one is the 4's series in the following patterns?

A \(3,\;6,\;9,\;12,....\)

B \(4,\;8,\;12,\;16,....\)

C \(1^3,\;2^3,\;3^3,\;4^3,\;...\)

D \(2,\;4,\;6,\;8,....\)

×

When the sequence of numbers follows a rule, it is called a number pattern.

We know that 4's series is defined by multiples of 4.

Multiples of 4,

\(4×1=4\\ 4×2=8\\ 4×3=12\\ 4×4=16\\ \vdots\\ \text {and so on}\)

Hence, option (B) is correct.

Which one is the 4's series in the following patterns?

A

\(3,\;6,\;9,\;12,....\)

.

B

\(4,\;8,\;12,\;16,....\)

C

\(1^3,\;2^3,\;3^3,\;4^3,\;...\)

D

\(2,\;4,\;6,\;8,....\)

Option B is Correct

Finding the Rule for a Given Pattern

  • Every pattern follows a specific rule.
  • Once we have a pattern, the rule for the pattern can be established.
  • There are many methods to find the rule for a given pattern.
  • Here, we are describing the simplest method.
  • We can write the rule for a given pattern using variables.

Consider an example:

\(3, \;6, \;9, \;12,......\)

In this example, we can use a variable to represent the terms in the given sequence.

Let \(x=\)  any term in the pattern

  • Here, the terms are \(3, \;6, \;9, \;12\) and so on.
  • On observing, we can say that every next term is \(3\) more than the previous term.
  • Since we are adding \(3\) to each term to get the next term, so we can say that \( x\) plus \(3\) is the rule.

\(x\) plus 3 means "\(x\)+3". 

  • Check the rule "\(x+3\)" to be sure that the rule is working.

               \(3, \;6, \;9, \;12,\dots\)

  • If we take \(3\) and put it for \(x\), the output is \(3+3=6\) 
  • If we take 6 and put it for \(x\), the output is  \(6+3=9\)
  • If we take \(9\) and put it for \(x\), the output is \(9+3=12\)

So, the rule is working.

Illustration Questions

\(3,\;5,\;9,\;17,\;33,.....\) Find the rule for the given pattern. ( \(x=\)  any term in the pattern)

A \(2x\)

B \(3x\)

C \(x+2\)

D \(2x-1\)

×

Checking all the four options to find out which rule is working.

Option (A) : \(2x\)

Here, \(x=\)  any term in the pattern

Putting the values for \(x\) from the given pattern,

\(2×3=6 \\ 2×5=10\\ 2×9=18\)

This rule does not work for the given pattern.

Hence, option (A) is incorrect.

Option (B) : \(3x\)

Putting the values for \(x\) from the given pattern,

\(3×3=9 \\ 3×5=15\\ 3×9=27\)

This rule does not work for the given pattern.

Hence, option (B) is incorrect.

Option (C) :  \(x+2\)

Putting the values for \(x\) from the given pattern,

\(3+2=5 \\ 5+2=7\\ 9+2=11\)

This rule does not work for the given pattern.

Hence, option (C) is incorrect.

Option (D) : \(2x-1\)

Putting the values for \(x\) from the given pattern,

\((2×3)-1=5 \\ (2×5)-1=9\\ (2×9)-1=17\\ (2×17)-1=33\\\)

This rule works for the given pattern.

Hence, option (D) is correct.

\(3,\;5,\;9,\;17,\;33,.....\) Find the rule for the given pattern. ( \(x=\)  any term in the pattern)

A

\(2x\)

.

B

\(3x\)

C

\(x+2\)

D

\(2x-1\)

Option D is Correct

Figure Pattern

In this type of pattern, there can be two changing features, first number of shapes or objects and other is types of shapes.

Consider the given example as shown.

Example 1: In this example, shapes are changing directions.

Example 2:

In this example, the number of shapes is changing.

  • By visual inspection, we can make a pattern from the given rule and vice versa.

Illustration Questions

A rule says that each circle follows a square. Which pattern is correct for the given rule?

A

B

C

D

×

We know that every pattern follows a rule.

The given rule says that each circle follows a square.

That means a circle comes after a square.

image

Hence, option (A) is correct.

A rule says that each circle follows a square. Which pattern is correct for the given rule?

A image
B image
C image
D image

Option A is Correct

Input-Output Table

  • A pattern is something that repeats in a specific way.
  • Once we have a pattern, a rule can be established for it.
  • A pattern can also be represented by an input-output table.
  • The input-output table shows the relation between input and output.
  • A term that has been put into the table is the input.
  • A term that comes out, is the output.
  • We can write a rule by examining the pattern in the input-output table.

How to write a rule from Input-Output table:

This can be done by following two steps:

(i) Think of the input as a variable.

(ii) Write the operations used with this variable to get the output.

Consider an example:

Input Output
0 0
1 4
2 8
3 12
  • By examining this pattern, we can say that the terms in the input column are multiplied by 4 to get the terms in the output column.
  • This is the rule for this table.
  • We can write the rule as an expression.
  • If the input column is \(x\) then  

Rule\(\rightarrow4x\)

Check the rule to be sure that it is working:

If we substitute \(0\) in place of \(x\), the output is: \(4×0=0\)

If we substitute \(1\) in place of \(x\), the output is: \(4×1=4\)

If we substitute \(2\) in place of \(x\), the output is: \(4×2=8\)

\(\vdots\)

So the rule is working.

 

  • The input-output can respectively be written as \(x\) and \(y\) in an input-output table.

  • We can solve many real world problems using the input-output table.

Illustration Questions

Write a rule for the given pattern: Input Output 1 3 2 4 3 5 4 6 5 7

A \(x+1\)

B \(x+2\)

C \(x+3\)

D \(x+4\)

×

We can observe that the terms in the output column are 2 more than the terms in the input column.

If the input column is \(x\) then we can write a rule for the given input-output table.

Rule \(\rightarrow x+2\)

Checking the rule to be sure that it is working:

If we substitute \(1\) in place of \(x\), the output is : \(1+2=3\)

If we substitute \(2\) in place of \(x\), the output is : \(2+2=4\)

If we substitute \(3\) in place of \(x\), the output is : \(3+2=5\)

\(\vdots\)

and so on.

So the rule is working.

Hence, option (B) is correct.

Write a rule for the given pattern: Input Output 1 3 2 4 3 5 4 6 5 7

A

\(x+1\)

.

B

\(x+2\)

C

\(x+3\)

D

\(x+4\)

Option B is Correct

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