Informative line

# General Number Series

• 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)

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 with the specific rule is called a number pattern.

Example: $$1,3,5,7,..... \\ 2,4,6,8,.....\\ 2,4,8,16,32,.....$$

### Figure Pattern:

In this type of pattern, there can be two changing features. First, the number of shapes or objects and types of shapes.

Example:  A list of numbers that follows a certain sequence with the specific rule is called a number pattern.

Generally, we come across many 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)

## Triangular Number Series

Triangular numbers are the sum of successive counting number starting from $$1.$$

$$1,\;3,\;6,\;10,\;15,\,...$$

$$\to$$ It can be defined by the following rule:

$$x_n=\dfrac{n(n+1)}{2}$$

Here, $$x_n=$$ value of $$n^{th}$$ term

$$n=$$ term which we want to find out

Example: Find out the $$15^{th}$$ term of the triangular number series.

We know triangular number series shows the sum of successive counting numbers starting from $$1.$$

To find out the $$15^{th}$$ term of the triangular number series, we use the following rule:

$$x_n=\dfrac{n(n+1)}{2}$$

Here, $$n=15$$

$$x_{15}=\dfrac{15(15+1)}{2}$$

$$=120$$

Thus, the $$15^{th}$$ term of the series is $$120$$.

## Square Number Pattern

Square numbers are the squares of successive counting numbers-

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

It can be defined by the following rule:

$$x_n=n^2$$

Here, $$x_n=$$ value of $$n^{th}$$  term; $$n=$$ term which we want to find out.

Example: Find the $$7^{th}$$ term of the square numbers series.

We know square number pattern shows the squares of successive counting numbers.

$$x_n=n^2$$

Here, $$n=7$$

$$\therefore\;7^{th}$$ term of the square number pattern is:

$$x_7=(7)^2=7×7=49$$

### Square Number Pattern

Square numbers are the squares of successive counting numbers-

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

It can be defined by the following rule:

$$x_n=n^2$$

Here, $$x_n=$$ value of $$n^{th}$$  term; $$n=$$ term which we want to find out.

Example: Find the $$7^{th}$$ term of the square numbers series.

We know square number pattern shows the squares of successive counting numbers.

$$x_n=n^2$$

Here, $$n=7$$

$$\therefore\;7^{th}$$ term of the square number pattern is:

$$x_7=(7)^2=7×7=49$$

#### What will be the $$10^{th}$$ term of the square number series?

A $$49$$

B $$100$$

C $$25$$

D $$36$$

×

Square number series follows the given rule:

$$x_n=n^2$$

Where $$n=$$ term to be found out

Here, $$n=10$$

So, $$n_{10}=(10)^2$$

$$=100$$

Hence, option (B) is correct.

### What will be the $$10^{th}$$ term of the square number series?

A

$$49$$

.

B

$$100$$

C

$$25$$

D

$$36$$

Option B is Correct

# Missing term of the Number Pattern

• The missing term of a number pattern can be found out by understanding the pattern rule.

Let's consider an example:

$$1,\;6,\;21,\;—,\;201$$

• In the given sequence, there is an increment. Thus, addition or multiplication rule is possible.
• First three terms of the sequence are $$1,\;6$$ and $$21$$.
• We can observe that there is a large increment in the numbers so we will try multiplication first.
• Let's try by multiplying with $$6.$$

$$1×6=6\\ 6×6=36$$

• This is not the exact sequence.

$$\to$$ Now, try by multiplying with $$3.$$

$$(1×3)=3\\ (6×3)=18$$

The product obtained are $$3$$ less than the corresponding terms of the given series.

$$\therefore$$ We add $$3$$ to the products obtained to get the exact terms of the series.

$$(1×3)+3=6\\ (6×3)+3=21$$

Thus, the pattern's rule is : Multiply by $$3$$ and then add $$3.$$

Let $$x=$$ any term of the pattern. So, the rule is: $$3x+3$$

Now, extend the pattern according to the rule.

$$(21×3)+3=66\\ (66×3)+3=201$$

Thus, the missing term is $$66$$.

#### Find the missing term of the given pattern:   $$9556,\;2388,\;596,\;—,\;36,\;8,\;1$$

A $$248$$

B $$184$$

C $$284$$

D $$148$$

×

In the given sequence, there is a decrement. Thus, subtraction or division rule is possible.

First three terms are $$9556,\;2388,\;596$$.

We can observe that there is a large decrement in numbers so we will try division first.

Let's try by dividing by $$4.$$

$$(9556/4)=2389\\ (2388/4)=597$$

The quotients obtained are $$1$$ more than the corresponding terms of the given series.

$$\therefore$$ We subtract $$1$$ from the quotients obtained to get the same series.

Thus, the rule is : First divide by $$4$$ and then subtract $$1.$$

Let $$x=$$ any term of the pattern so, the rule is : $$\dfrac{x}{4}-1$$

$$(9556/4)-1=2388\\ (2388/4)-1=596\\ (596/4)-1=148\\ (148/4)-1=36$$

Thus, the missing term is $$'148'$$.

Hence, option (D) is correct.

### Find the missing term of the given pattern:   $$9556,\;2388,\;596,\;—,\;36,\;8,\;1$$

A

$$248$$

.

B

$$184$$

C

$$284$$

D

$$148$$

Option D is Correct

# Geometric Pattern

A sequence of repetitive geometric figures or changing shapes or changing directions, with a specific rule is called a geometric pattern.

For example:

In the given pattern, we can see multiple geometric figures are repeating in groups. Two triangles are followed by a circle which is then followed by a rectangle.  #### What is the sequence of the given geometric pattern?

A Sequence of triangles rotating in anti-clockwise direction.

B Sequence of triangles rotating in clockwise direction.

C Not following any sequence.

D Sequence of squares rotating in clockwise direction.

×

Given geometric pattern is : In the given pattern, we can see triangles are pointing in different directions, which shows a geometric pattern.

We can observe that there is a sequence of triangles rotating in anti-clockwise direction.

Hence, option (A) is correct.

### What is the sequence of the given geometric pattern? A

Sequence of triangles rotating in anti-clockwise direction.

.

B

Sequence of triangles rotating in clockwise direction.

C

Not following any sequence.

D

Sequence of squares rotating in clockwise direction.

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.

#### 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

# Fibonacci Series

• Fibonacci series is a set of numbers starting from zero and each successive number is the sum of the preceding two numbers.  $$\to$$ We can write a rule for the Fibonacci series.

$$x_n=x_{n-1}+x_{n-2}$$

Here, $$x_n=$$ term in the series

$$x_{n-1}=$$ preceding the first number

$$x_{n-2}=$$ preceding the second number

#### Which one of the following is a Fibonacci series?

A $$1,\;2,\;3,\;5,\;9$$

B $$0,\;1,\;1,\;2,\;3,\;5$$

C $$1,\;3,\;5,\;7$$

D $$1,\;1,\;2,\;2,\;3$$

×

The basic rule of Fibonacci series is that each number is the sum of the preceding two numbers.

In option (A), the last term $$(9)$$ is not equal to the sum of its two preceding numbers $$(3,\;5)$$.

$$3+5\neq9$$

Hence, option (A) is incorrect.

In option (B), the last term $$(5)$$ is equal to the sum of its two preceding numbers $$(3,\;2)$$.

$$2+3=5$$

Now, we check all the numbers by Fibonacci's rule.

$$0+1=1\\ 1+1=2\\ 2+1=3\\ 2+3=5$$

Hence, option (B) is correct.

In option (C), the last term $$(7)$$ is not equal to the sum of its two preceding numbers $$(3,\;5)$$.

$$3+5\neq7$$

Hence, option (C) is incorrect.

In option (D), the last term $$(3)$$ is not equal to the sum of its two preceding numbers $$(2,\;2)$$

$$2+2\neq3$$

Hence, option (D) is incorrect.

### Which one of the following is a Fibonacci series?

A

$$1,\;2,\;3,\;5,\;9$$

.

B

$$0,\;1,\;1,\;2,\;3,\;5$$

C

$$1,\;3,\;5,\;7$$

D

$$1,\;1,\;2,\;2,\;3$$

Option B is Correct

# Finding Rule for Geometric Patterns

## Geometric pattern

• A sequence of repetitive geometric figures or changing shapes or changing directions, with a specific rule is called a geometric pattern.
• We can write a rule for geometric patterns also.
• Follow the given steps to find a rule for geometric patterns.

Step 1: Analyze the geometric pattern and write it's sequence as a number series.

Step 2: Now write a rule for this series like a number pattern.

• Consider an example to understand it better.  $$\to$$ Here, each term has a different number of units (squares).

$$\to$$ Write it's sequence as a number series.

i.e., $$1,\;3,\;5,\;7\,,...$$

• Since, square of one is common in all terms,

$$\to$$ So, the pattern can be written as;

$$1+0,\;1+2,\;1+4,\;1+6,\,...$$

• Each successive term is increasing with the multiples of $$2.$$

$$\to$$ So, the rule can be : $$1+2n$$.

$$\to$$ Let's check for the above pattern,

$$1+2(1)=1+2=3$$

Which is not the $$1^{st}$$ term.

Try with another rule,

$$1+2(n-1)$$

$$\Rightarrow\;1+2n-2$$

$$\Rightarrow\;2n-1$$

Let's check the pattern with this rule,

$$2(1)-1=2-1=1\\ 2(2)-1=4-1=3\\ 2(3)-1=6-1=5$$

Thus, the rule for the given geometric pattern is $$(2n-1)$$

where $$n$$ represents the term which is to be found out.

#### Find a rule for the given geometric pattern:

A $$4n-3$$

B $$2n-3$$

C $$4n+1$$

D $$4n-1$$

×

Each term of the given geometric pattern has a different number of units (squares).

Now, write the sequence as a number series.

$$1,\;5,\;9,\;13\,,...$$

Since, square of one is common is all the terms,

so, the pattern can be written as-

$$1+0,\;1+4,\;1+8,\;1+12,\,...$$

Each successive term is increasing with the multiples of $$4.$$

So, the rule can be : $$1+4n$$

Checking of rule for the above pattern:

$$1+4×1=1+4=5$$

Which is not the $$1^{st}$$ term.

Let's try with another rule,

$$1+4(n-1)$$

$$=1+4n-4$$

$$=4n-3$$

Check the pattern with this rule,

$$4(1)-3=4-3=1\\ 4(2)-3=8-3=5\\ 4(3)-3=12-3=9\\ 4(4)-3=16-3=13$$

Thus, the rule for the given geometric pattern is $$(4n-3)$$, where $$n$$ represents the term which is to be found out.

Hence, option (A) is correct.

### Find a rule for the given geometric pattern: A

$$4n-3$$

.

B

$$2n-3$$

C

$$4n+1$$

D

$$4n-1$$

Option A is Correct

# Number Pattern Involving Combination of Two Operations

• A number pattern follows a specific rule either of a single operation or of multiple operations.

For example: $$1,\;3,\;5,\;7\,...$$ is a number pattern that follows a specific rule by of $$'+2'$$ i.e. addition operation.

• Sometimes specific rules of single operation do not work.

For example: $$1,\;4,\;19,\;94\,...$$

$$\to$$ This number pattern is not following a specific rule of single operation.

• To identify the rule for this type of number pattern, we should apply the combination of two operations.
• There are four combinations of operations, having two operations each. These are as follows:

• This combination works for increasing-number patterns.
• When increment in the consecutive terms of a series is more than the multiples of a number, this combination is used.

(2) Multiplication with subtraction:

• This combination works for increasing-number patterns.
• When increment in the consecutive terms of a series is less than the multiples of a number, this combination is used.

• This combination works for decreasing number patterns.
• When decrement in the consecutive terms of a series is more than the multiples of a number, this combination is used.

(4) Division with subtraction:

• This combination works for decreasing number patterns.
• When decrement in the consecutive terms of a series is less than the multiples of a number, this combination is used.
• Let's consider our previous example again.

$$1,\;4,\;19,\;94\,...$$

$$\to$$ Since, this is an increasing-number pattern therefore, multiplication with addition or multiplication with subtraction combination should work.

$$\to$$ When we multiply each term by $$5$$ we get,

$$\begin{array}c 1×5&=&5\\ 4×5&=&20\\ 19×5&=&95\\ \end{array}$$

and so on.

$$\to$$ We can observe that products obtained are $$1$$ more than the corresponding terms of the given series.

Thus, multiplication with subtraction combination is to be used.

$$\therefore$$ Subtract $$1$$ from each term which is obtained by multiplying by $$5.$$

$$\begin{array}c (1×5)-1&=&5-1&=&4\\ (4×5)-1&=&20-1&=&19\\ (19×5)-1&=&95-1&=&94\\ \end{array}$$

and so on.

• So, the rule is: Multiply by $$5$$ and subtract $$1.$$
• Next term of this number pattern using above rule is-

$$(94×5)-1=470-1=469$$

#### Find the next term of the following number pattern: $$802,\;402,\;202,\;102,\,...$$

A $$51$$

B $$50$$

C $$52$$

D $$53$$

×

Given number pattern:

$$802,\;402,\;202,\;102,\,...$$

Since, this is a decreasing-number pattern therefore division with addition or division with subtraction combination should work.

Divide each term by $$2$$ we get,

$$\begin{array}c 802\div2&=&401\\ 402\div2&=&201\\ 202\div2&=&101\\ \end{array}$$

We can observe that the quotients obtained are $$1$$ less than the corresponding terms of the given series.

Thus, division with addition combination is to be used.

$$\therefore$$ Add $$1$$ to each term which is obtained by dividing by $$2.$$

$$\begin{array}c (802\div2)+1&=&401+1=402\\ (402\div2)+1&=&201+1=202\\ (202\div2)+1&=&101+1=102\\ \end{array}$$

So, the rule is: Divide by $$2$$ and add $$1.$$

The next term using the above rule is: $$(102\div2)+1=51+1=52$$

Hence, option (C) is correct.

### Find the next term of the following number pattern: $$802,\;402,\;202,\;102,\,...$$

A

$$51$$

.

B

$$50$$

C

$$52$$

D

$$53$$

Option C is Correct

# Patterns Involving Single Arithmetic Operations

• Numbers form various patterns.
• Infinite patterns can be made using different operations.
• We know that every number pattern is defined by a rule.
• Here, we will consider the patterns involving some basic operations.

When the consecutive terms in a sequence increase by a small constant value, we can use the addition rule to find more terms in that pattern.

Ex. $$1, \;3, \;5, \;7, \;9,......$$

## Subtraction Rule

When the consecutive terms in a sequence decrease by a small constant value, we can use the subtraction rule to find more terms in that pattern.

Ex. $$16, \;14, \;12, \;10, ...$$

## Multiplication Rule

When the increase in consecutive terms of a sequence is in multiples, we can use the multiplication rule to find more terms in that pattern.

Ex. $$3, \;6, \;12, \;24, ...$$

## Division Rule

When the decrease in consecutive terms of a sequence is in multiples, we can use the division rule to find more terms in that pattern.

Ex. $$120, \;60, \;30, \; ...$$

#### Choose the next number in the given pattern: $$20, \;17, \;14, \;11,......$$

A $$7$$

B $$8$$

C $$9$$

D $$10$$

×

The consecutive numbers in the given sequence decrease by a small constant value, so it follows the subtraction rule.

First three terms of the given pattern are $$20,\;17,$$ and $$14$$.

Let $$x$$ = any term in the pattern

Since we are subtracting $$3$$ from each term to get the next term, so the rule is $$x$$ minus 3.

Rule $$\rightarrow x-3$$

First term = $$20$$

By subtracting $$3$$ from every term, we get:

$$20-3=17$$ Second term

$$17-3=14$$ Third term

$$14-3=11$$ Fourth term

$$11-3=8$$ Fifth term

Hence, option (B) is correct.

### Choose the next number in the given pattern: $$20, \;17, \;14, \;11,......$$

A

$$7$$

.

B

$$8$$

C

$$9$$

D

$$10$$

Option B is Correct