Informative line

Finding Terms Of A Pattern

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. \(120, \;60, \;30, \; ...\)

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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:

 \(2^5=2×2×2×2×2=32\)

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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 :

image
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.

Illustration Questions

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:

image
A

\(49\)

.

B

\(10\)

C

\(12\)

D

\(16\)

Option D is Correct

Practice Now