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Properties of Square - Based on Pattern

Square and Square Root I - Concepts

Perfect Square

Square of Number 1-50

Square Root Using Prime Factorization

General Properties of Squares

Properties of Square - Based on Ending Digits

Properties of Square - Unit Digit of Square

Properties of Square - Difference of Consecutive Numbers

Pythagorean Triplet

Number Between Square Numbers

Properties of Square - Based on Pattern

Class - 8th CBSE Subjects

Concept Explanation

Properties of Square - Based on Pattern

A pattern refers to a series or sequence which repeats. There are two major kinds of math patterns. They are number patterns or sequences of numbers. These are arranged as per a rule or rules or a formula. The other type we have are shape patterns that are tagged by making use of letters in a definite shape and the way that they repeat.

Some interesting patterns of square numbers :

Pattern 1:The sum of two consecutive triangular numbers is a square number .

Numbers whose dot patterns can be arranged as triangles are known as triangular numbers.1,3,6,10,15,21,... are triangular numbers.This fact can be observed by the following pattern

$1=1^{2}$

$1+3=4=2^{2}$

$3+6=9=3^{2}$

$6+10=16=4^{2}$

$10+15=25=5^{2}$

$15+21=36=6^{2}$ etc.

Also n^th triangular number is given by $frac{n(n+1)}{2}$

Pattern 2: If 1 is added to the product of two consecutive odd natural numbers,it is equal to the square of the only even natural number between them.

This fact can be observed by the following pattern

$1times3+1=4=2^{2}$

$3times5+1=16=4^{2}$

$5times7+1=36=6^{2}$

$7times9+1=64=8^{2}$

$9times11+1=100=(10)^{2}$ etc.

In general, $(2n-1)times(2n+1)+1=4n^{2}=(2n)^{2}$

Pattern 3: If 1 is added to the product of two consecutive even natural numbers,it is equal to the square of the only odd natural number between them.

This fact can be observed by the following pattern

$2times4+1=9=3^{2}$

$4times6+1=25=5^{2}$

$6times8+1=49=7^{2}$

$8times10+1=81=9^{2}$

$10times12+1=121=(11)^{2}$ etc.

In general: $2ntimes(2n+2)+1=4n^{2}+4n+1=(2n+1)^{2}$

Pattern 4:The square of any odd natural number other than 1 can be expressed as the sum of two consecutive natural numbers.

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Sample Questions

(More Questions for each concept available in Login)

Question : 1

Using the given pattern,find the missing numbers: $1^{2}+2^{2}+2^{2}=3^{2}$ $2^{2}+3^{2}+6^{2}=7^{2}$ $3^{2}+4^{2}+(12)^{2}=13^{2}$ $4^{2}+5^{2}+(.....)^{2}=(21)^{2}$ $5^{2}+(......)^{2}+(30)^{2}=(31)^{2}$ $6^{2}+(7)^{2}+(.....)^{2}=(.....)^{2}$
A. 20,6,43,48	B. 20,6,42,43	C. 21,25,43,42	D. none of these
Right Option : B
View Explanation

Question : 2

Observe the pattern $2^{2}-1^{2}=2+1$ $3^{2}-2^{2}=3+2$ $4^{2}-3^{2}=4+3$ $5^{2}-4^{2}=5+4$ find $(99)^{2}-(96)^{2}$
A. 584	B. 585	C. 589	D. 564
Right Option : B
View Explanation

Question : 3

Express $(21)^{2}$ as the sum of two consecutive natural numbers.
A. 220+221	B. 100+341	C. 400+41	D. none of these
Right Option : A
View Explanation

Chapters

Rational Numbers I

Rational Numbers II

Linear Equations in One Variable I

Square and Square Root I

Square and Square Root II

Cube and Cube Roots

Comparing Quantities I

Comparing Quantities II

Comparing Quantities III

Algebraic Expressions

Algebraic Identities

Mensuration I

Mensuration II

Exponents and Power I

Exponents and Power II

Direct and Inverse Variation I

Direct and Inverse Variation II

Factorisation

Introduction to Graphs

Constructions

Prerequiste

Playing With Numbers

Linear Equations in One Variable II

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Attention !!!!!

Properties of Square - Based on Pattern

Properties of Square - Based on Pattern

Some interesting patterns of square numbers :

Pattern 1:The sum of two consecutive triangular numbers is a square number .

Pattern 2: If 1 is added to the product of two consecutive odd natural numbers,it is equal to the square of the only even natural number between them.

Pattern 3: If 1 is added to the product of two consecutive even natural numbers,it is equal to the square of the only odd natural number between them.

Pattern 4:The square of any odd natural number other than 1 can be expressed as the sum of two consecutive natural numbers.

Students / Parents Reviews [10]

Eshan Arora

Cheshta

Om Umang

Shivam Rana

Prisha Gupta

Shreya Shrivastava

Yatharthi Sharma

Hiya Gupta

Aman Kumar Shrivastava

Barkha Arora