Properties of Square - Based on Pattern

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.

.... (More Text Available, Login?)

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

Observe the pattern $2^{2}-1^{2}=2+1$ $3^{2}-2^{2}=3+2$ $4^{2}-3^{2}=4+3$ $5^{2}-4^{2}=5+4$ and find $(111)^{2}-(109)^{2}$
A. 420	B. 430	C. 340	D. 440
Right Option : D
View Explanation

Chapters

Time, Speed and Distance

Problems On Boats And Stream

Problems On Trains

Pipe And Cistern

Simplification

Rules For Divisibility

Number Theory

Cube and Cube Roots

Square and Square Root

Permutation and Combination

Partnership and Mixture

Simple and Compound Interest

Profit Loss and Discount

Percentages

Fractions and Decimals

Content / Category

IMO-Mathematics Olympiad

Central Government Service

ISO - Science Olympiads

State Government Services

Banking And Insurance

Railways & Metro Services

MBA Entrance

Law Entrance

English Proficiency

Reasoning Proficiency

Computer Proficiency

Navodaya Entrance 6th & 9th

Sainik School Entrance

Andhra Bank

Class / Course

IBPS PO Prelims

IBPS Clerk Prelims

IBPS PO Prelims Test Series

SBI SO General Aptitude

Bank PO Prelims

Bank Clerical Prelims

Bank AO Prelims

Bank SO Prelims

Quantitative Aptitude Proficiency

Verbal Ability Proficiency

Bank Of Baroda AO Prelims

LIC ADO Prelims

LIC AAO Prelims

IBPS Clerk Prelims Test Series

IBPS RRB Prelims Test Series

IBPS SO Prelims Test Series

SBI PO Prelims Test Series

SBI Clerical Prelims Test Series

SBI SO General Aptitude Test Series

Bank PO Prelims Test Series

Bank Clerical Prelims Test Series

Bank AO Prelims Test Series

Bank SO Prelims Test Series

Quantitative Aptitude Test Series

EXIM AO Prelims Test Series

Verbal Ability Test Series

Bank Of Baroda AO Prelims Test Series

LIC ADO Prelims Test Series

LIC AAO Prelims Test Series

Why ABHYAS?

Self Learning

Course Benefits

Our Areas of Expertise

IMO-Mathematics Olympiad

Central Government Service

ISO - Science Olympiads

State Government Services

Banking And Insurance

Railways & Metro Services

MBA Entrance

Law Entrance

English Proficiency

Reasoning Proficiency

Computer Proficiency

Navodaya Entrance 6th & 9th

Sainik School Entrance

Andhra Bank

Students / Parents Reviews [10]

Abhyas is a complete education Institute. Here extreme care is taken by teacher with the help of regular exam. Extra classes also conducted by the institute, if the student is weak.

Om Umang

10th

It has a great methodology. Students here can get analysis to their test quickly.We can learn easily through PPTs and the testing methods are good. We know that where we have to practice

Barkha Arora

10th

One of the best institutes to develope a child interest in studies.Provides SST and English knowledge also unlike other institutes. Teachers are co operative and friendly online tests andPPT develo...

Aman Kumar Shrivastava

10th

I have spent a wonderful time in Abhyas academy. It has made my reasoning more apt, English more stronger and Maths an interesting subject for me. It has given me a habbit of self studying

Yatharthi Sharma

10th

My experience with Abhyas is very good. I have learnt many things here like vedic maths and reasoning also. Teachers here first take our doubts and then there are assignments to verify our weak poi...

Shivam Rana

7th

It was a good experience with Abhyas Academy. I even faced problems in starting but slowly and steadily overcomed. Especially reasoning classes helped me a lot.

Cheshta

10th

About Abhyas metholodology the teachers are very nice and hardworking toward students.The Centre Head Mrs Anu Sethi is also a brilliant teacher.Abhyas has taught me how to overcome problems and has...

Shreya Shrivastava

8th

Abhyas Methodology is very good. It is based on according to student and each child manages accordingly to its properly. Methodology has improved the abilities of students to shine them in future.

Manish Kumar

10th

My experience with Abhyas academy is very good. I did not think that my every subject coming here will be so strong. The main thing is that the online tests had made me learn here more things.

Hiya Gupta

8th

A marvelous experience with Abhyas. I am glad to share that my ward has achieved more than enough at the Ambala ABHYAS centre. Years have passed on and more and more he has gained. May the centre f...

Archit Segal

7th

View All

Attention !!!!!