Informative line

# 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.

## Addition Rule

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

# Patterns Involving Combination of Multiplication and Addition

• In various patterns we will find that a single rule is not applicable, so we need to use the combination of two rules.
• Here, we will discuss those patterns where the rule is a combination of multiplication and addition.

For example:

Consider the following sequence:

$$5, \;11, \;23, \;47,....$$

• In the above sequence, there is an increase in each term, but that increase can not be defined by a single rule.
• The increase is in multiples but not in exact multiples.

Let's check it.

• First three terms of the given sequence are $$5,\,11$$ and $$23$$.

Let $$x=$$ any term of the pattern

• Let's try by multiplying each term by $$2$$, i.e. $$2x$$.
• Start with $$5$$.

$$5×2=10\\ 11×2=22\\ 23×2=46$$

• The products are not same as the given sequence.
• We can observe that the products obtained are $$1$$ less than the corresponding terms of the given sequence.
• So, multiply each term by $$2$$ and then add $$1$$ to it, to obtain the next term.

$$(5×2)+1=11\\ (11×2)+1=23\\ (23×2)+1=47\\ \vdots$$

and so on.

Now, it is same as the given sequence.

So, the rule is  $$2x+1$$

• We can write the rule for this sequence as:

Multiply each term by $$2$$  and add $$1$$.

• There would be different rules for every different sequence.

#### A number pattern is shown: $$7,\;16,\;34,\;70,...$$ Find the next term in the pattern.

A $$138$$

B $$142$$

C $$130$$

D $$140$$

×

$$7,\;16,\;34,\;70,....$$

In the given pattern, each term is increasing by a large difference, like in case of multiples.

Let $$x =$$ any term of the pattern

First three terms of the given pattern are $$7,\;16$$ and $$34$$.

Let's try by multiplying each term by $$2$$, starting with $$7$$.

$$7×2=14\\ 16×2=32\\ 34×2=68\\$$

But it is not same as the given sequence. It means the single rule is not defined here.

We can observe that the products obtained are $$2$$ less than the corresponding terms of the given sequence.

So, we multiply each term by $$2$$ and then add $$2$$ to it, to obtain the next term.

Thus, the rule will be $$2x+2$$.

First term = $$7$$

Second term =$$(7×2)+2=16$$

Third term = $$(16×2)+2=34$$

Fourth term = $$(34×2)+2=70$$

Fifth term = $$(70×2)+2=142$$

$$142$$ is the next term of the given pattern.

Hence, option (B) is correct.

### A number pattern is shown: $$7,\;16,\;34,\;70,...$$ Find the next term in the pattern.

A

$$138$$

.

B

$$142$$

C

$$130$$

D

$$140$$

Option B is Correct

# Patterns Involving Combination of Division and Subtraction

• Here, we will discuss those patterns where the rule is a combination of division and subtraction.

For example:  Consider the following sequence:

$$382,\;190,\;94,\;46,...$$

• In the above sequence, there is a decrease in each term, but that decrease can not be defined by a single rule.
• The decrease is in multiples but not in exact multiples.
• So there is a chance of division operation.

Let's check it.

Let $$x =$$  any term in the pattern

• First three terms of the given sequence are $$382,\;190$$ and $$94$$.
• Let's try by dividing each term by $$2$$, starting with $$382$$.

$$\dfrac {382}{2}=191$$

$$\dfrac {190}{2}=95$$

• But it is not same as the given sequence.
• We can observe that the quotients obtained are  $$1$$ more than the corresponding terms of the given sequence.
• This means a single rule is not defined here.
• Now, we will divide each term by $$2$$ and then subtract $$1$$ from it to obtain the next term.

Rule: $$\dfrac {x}{2}-1$$

$$\text {First Term}\rightarrow382$$

$$\dfrac {382}{2}-1=191-1=190\leftarrow\;\text {Second Term}$$

$$\dfrac {190}{2}-1=95-1=94\leftarrow\;\text {Third Term}$$

$$\dfrac {94}{2}-1=47-1=46\leftarrow\;\text {Fourth Term}$$

Now, it is same as the given sequence.

• We can write the rule for this sequence as:

Divide each term by $$2$$ and then subtract $$1$$ from it.

• There would be different rules for every different sequence.

#### A number pattern is shown: $$76,\;36,\;16,\,...,\;1$$ Find the missing term in the given pattern.

A $$6$$

B $$8$$

C $$10$$

D $$12$$

×

$$76,\;36,\;16,...$$

In the given pattern, each term is decreasing by a large difference, like in case of multiples.

Let $$x =$$  any term of the pattern

First three terms of the given pattern are $$76,\;36$$ and $$16$$.

Let's try by dividing each term by $$2$$, starting with $$76$$.

$$\dfrac {76}{2}=38$$

$$\dfrac {36}{2}=18$$

But it is not same as the given sequence. This means a single rule is not defined here.

We can observe that the quotients obtained are $$2$$ more than the corresponding terms of the given sequence.

So, we will divide each term by $$2$$ and then subtract $$2$$ from it to obtain the next term.

Rule: $$\dfrac {x}{2}-2$$

The terms are:

$$\text {First Term}=76$$

$$\dfrac {76}{2}-2=38-2=36\,\leftarrow\text{Second Term}$$

$$\dfrac {36}{2}-2=18-2=16\,\leftarrow\text{Third Term}$$

$$\dfrac {16}{2}-2=8-2=6\,\leftarrow\text{Fourth Term}$$

$$\dfrac {6}{2}-2=3-2=1\,\leftarrow\text{Fifth Term}$$

$$\therefore$$ The required missing term is $$6$$.

Hence, option (A) is correct.

### A number pattern is shown: $$76,\;36,\;16,\,...,\;1$$ Find the missing term in the given pattern.

A

$$6$$

.

B

$$8$$

C

$$10$$

D

$$12$$

Option A is Correct

# Patterns Involving Powers

## Power Pattern:

The number patterns in which the increase is in powers are called power patterns.

• In a pattern, if the increase is only in powers, and the base is same of all the terms, we can obtain more terms in the sequence by increasing the powers.

For example: Consider the following sequence:

$$2,\,4,\,8,\,16............$$

Now we have to find the next term in the given sequence.

We know,

$$2=2^1 \\ 4=2^2\\8= 2^3\\ 16=2^4$$

• We can observe that in this pattern, the base is same of all the terms.
• The increase is only in powers.
• So, the next term of the given sequence is:

$$2^5=2×2×2×2×2=32$$

#### A number pattern is shown: $$3,\,9,\,27..........$$ Find the next term of the given pattern.

A $$81$$

B $$12$$

C $$51$$

D $$72$$

×

Given: $$3,\,9,\,27.........$$

We know that the above sequence can be written as:

$$3=3^1\\9=3^2\\27=3^3$$

We can observe that the power is increasing in each term, but the base is same.

Thus, we can get the next term by increasing the power.

$$\therefore$$ $$3^1=3$$                                   First Term

$$3^2=3×3=9$$                       Second Term

$$3^3=3×3×3=27$$             Third Term

$$3^4=3×3×3×3=81$$      Fourth Term

Hence, option (A) is correct.

### A number pattern is shown: $$3,\,9,\,27..........$$ Find the next term of the given pattern.

A

$$81$$

.

B

$$12$$

C

$$51$$

D

$$72$$

Option A is Correct

# Patterns Involving Combination of Multiplication and Subtraction

• In various patterns we will find that a single rule is not applicable, so we need to use the combination of two rules.
• Here, we will discuss those patterns where the rule is a combination of multiplication and subtraction.

Example:

Consider the following sequence:

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

• In the above sequence, there is an increase in each term, but that increase can not be defined by a single rule.
• The increase is in multiples but not in exact multiples.
• So there is a chance of multiplication operation.

Let's check it.

First three terms of the given sequence are $$1$$$$4$$ and $$19$$.

Let's try by multiplying each term by 5, starting with 1.

$$1×5=5\\ 4×5=20\\ 19×5=95\\ \vdots$$

But it is not same as the given sequence.

We can observe that the products obtained are 1 more than the corresponding terms of the given sequence.

So, multiply each term by 5 and then subtract 1 from it to obtain the next term.

$$(1×5)-1=4\\ (4×5)-1=19\\ (19×5)-1=94\\ \vdots$$

and so on.

Now, it is same as the given sequence.

• We can write the rule for this sequence as:

Multiply each term by $$5$$ and subtract $$1$$ from it.

• There would be different rules for every different sequence.

#### A number pattern is shown: $$2,\;7,\;27,\;107,...$$ Find the next term in the pattern.

A $$344$$

B $$270$$

C $$184$$

D $$427$$

×

$$2,\;7,\;27,\;107,...$$

In the given pattern, each term is increasing by a large difference, like in case of multiples.

First three terms of the given pattern are $$2,\,7$$ and $$27$$.

Let's try by multiplying each term by $$4$$, starting with $$2$$.

$$2×4=8\\ 7×4=28\\ 27×4=108\\ \vdots$$

But it is not same as the given sequence. This means a single rule is not defined here.

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

So, we multiply each term by $$4$$ and then subtract $$1$$ from it, to obtain the next term.

$$(2×4)-1=7\\ (7×4)-1=27\\ (27×4)-1=107\\ (107×4)-1=427\\ \vdots$$

Next term of the given pattern is $$427$$.

Hence, option (D) is correct.

### A number pattern is shown: $$2,\;7,\;27,\;107,...$$ Find the next term in the pattern.

A

$$344$$

.

B

$$270$$

C

$$184$$

D

$$427$$

Option D is Correct

# Patterns Involving Combination of Division and Addition

• Here, we will discuss those patterns where the rule is a combination of division and addition.

For example: Consider the following sequence:

$$164,\;84,\;44,\;24,...$$

• In the above sequence, there is a decrease in each term, but that decrease can not be defined by a single rule.
• The decrease is in multiples but not in exact multiples.
• So there is a chance of division operation.

Let's check it.

Let $$x =$$ any term in the pattern

• First three terms of the given sequence are $$164,\;84$$ and $$44$$.
• Let's try by dividing each term by $$2$$, starting with $$164$$.

$$\dfrac {164}{2}=82$$

$$\dfrac {84}{2}=42$$

$$\dfrac {44}{2}=22$$

• But it is not same as the given sequence.
• We can observe that the quotients obtained are 2 less than the corresponding terms of the given sequence.
• So, divide each term by 2 and then add 2 to it to obtain the next term.

Rule: $$\dfrac {x}{2}+2$$

$$\dfrac {164}{2}+2=84$$

$$\dfrac {84}{2}+2=44$$

$$\dfrac {44}{2}+2=24$$

and so on.

Now, it is same as the given sequence.

• We can write the rule for this sequence as:

Divide each term by $$2$$ and then add $$2$$ to it.

• There would be different rules for every different sequence.

#### A number pattern is shown: $$242,\;122, \;62,..,17$$ Find the missing term in the given pattern.

A $$51$$

B $$34$$

C $$32$$

D $$22$$

×

$$242,\;122, \;62,....$$

In the given pattern, each term is decreasing by a large difference, like in case of multiples.

Let $$x =$$ any term in the pattern

First three terms of the given pattern are $$242,\;122$$ and $$62$$.

Let's try by dividing each term by $$2$$, starting with $$242$$.

$$\dfrac {242}{2}=121$$

$$\dfrac {122}{2}=61$$

But it is not same as the given sequence. This means a single rule is not defined here.

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

So, we divide each term by $$2$$ and then add $$1$$ to it to obtain the next term.

Rule: $$\dfrac {x}{2}+1$$

The terms are:

$$\text {First Term}\rightarrow242$$

$$\dfrac {242}{2}+1=121+1=122\leftarrow\text {Second Term}$$

$$\dfrac {122}{2}+1=61+1=62\leftarrow\text {Third Term}$$

$$\dfrac {62}{2}+1=31+1=32\leftarrow\text {Fourth Term}$$

$$\dfrac {32}{2}+1=16+1=17\leftarrow\text {FifthTerm}$$

$$\therefore$$ The required missing term is $$32$$.

Hence, option (C) is correct.

### A number pattern is shown: $$242,\;122, \;62,..,17$$ Find the missing term in the given pattern.

A

$$51$$

.

B

$$34$$

C

$$32$$

D

$$22$$

Option C is Correct

# Triangular Patterns

## Triangular numbers:

• Triangular numbers are the sum of successive counting numbers starting with $$1$$.

• These triangular numbers can be represented as triangular series.
• The sequence is generated from a pattern of dots, blocks, or any shape which forms a triangle.

Following is the triangular number sequence:

$$1,\;3,\;6,\;10,\;15,\;21,\;28,...$$

This can be represented as shown in the figure.  We can write a rule for the triangular numbers.

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

Here, $$x_n$$ is the value of the $$n^{th}$$ term.

$$n$$ is the number of terms.

Put $$n=1$$ then $$x_1=\dfrac {1(1+1)}{2}=\dfrac {2}{2}=1$$

Put $$n=2$$ then $$x_2=\dfrac {2(2+1)}{2}=\dfrac {6}{2}=3$$

Put $$n=3$$ then $$x_3=\dfrac {3(3+1)}{2}=\dfrac {12}{2}=6$$

and so on.

• This shows our rule is working.
• On putting the value of $$n$$, we can get any term of the given pattern.

#### Choose the $$5^{th}$$term of the given triangular pattern :

A $$12$$

B $$13$$

C $$10$$

D $$15$$

×

We know that a triangular pattern shows the sum of successive counting numbers starting with $$1$$.

Sum of successive counting numbers:

$$\begin{array} {} \text {First Term} & 1=1 \\ \text {Second Term} & 1+2=3 \\ \text {Third Term} & 1+2+3=6 \\ \text {Fourth Term} & 1+2+3+4=10 \\ \text {Fifth Term} & 1+2+3+4+5=15 \\ \end{array}$$

Hence, option (D) is correct.

### Choose the $$5^{th}$$term of the given triangular pattern : A

$$12$$

.

B

$$13$$

C

$$10$$

D

$$15$$

Option D is Correct

# Square Patterns

• A square pattern represents the squares of successive counting numbers.
• The pattern forms a square, so it is called square pattern.
• The square numbers can be represented as square groups.
• The sequence is generated from a pattern of dots which forms a square.

Following is the square number sequence:

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

This can be represented as shown.  • In a square pattern, the number of rows is equal to the number of columns.
• When the number of rows is not equal to the number of columns, then it is a rectangular pattern.

#### Choose the next term of the given square pattern:

A $$49$$

B $$10$$

C $$12$$

D $$16$$

×

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

Squares of successive counting numbers,

$$\text {First Term}\rightarrow 1^2=1$$

$$\text {Second Term}\rightarrow 2^2=4$$

$$\text {Third Term}\rightarrow 3^2=9$$

$$\text {Fourth Term}\rightarrow 4^2=16$$

Hence, option (D) is correct.

### Choose the next term of the given square pattern: A

$$49$$

.

B

$$10$$

C

$$12$$

D

$$16$$

Option D is Correct