Informative line

Patterns And Series

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:

\(x_7=(7)^2=7×7=49\)

 

 

Illustration Questions

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 \(66\).

Illustration Questions

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.

Illustration Questions

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 :

image

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?

image
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.

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

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

Illustration Questions

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.

Illustration Questions

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:

image
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:

(1) Multiplication with addition:

  • 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.

(3) Division with addition:

  • 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- 

\((94×5)-1=470-1=469\)

Illustration Questions

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.

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

Practice Now