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

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

Question : 2

Observe the following 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 the value of $(100)^{2}-(99)^{2}$
A. 99	B. 1	C. 199	D. 299
Right Option : C
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

Data Interpretation

Problems On Boats And Stream

Problems On Trains

Pipe And Cistern

Time, Speed and Distance

Quadratic Equations

Algebra

Simplification

Rules For Divisibility

Number Theory

Cube and Cube Roots

Square and Square Root

Permutation and Combination

Partnership and Mixture

Surface Area and Volume

Simple and Compound Interest

Profit Loss and Discount

Content / Category

IMO-Mathematics Olympiad

Central Government Service

ISO - Science Olympiads

State Government Services

Banking And Insurance

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

MBA Entrance

Law Entrance

English Proficiency

Reasoning Proficiency

Computer Proficiency

Navodaya Entrance 6th & 9th

Sainik School Entrance

Andhra Bank

Students / Parents Reviews [20]

We started with lot of hope that Abhyas will help in better understnding of complex topics of highers classes. we are not disappointed with the progress our child has made after attending Abhyas. Though need to mention that we expected a lot more. On a scale of 1-10, we would give may be 7.

Manya

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

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

Usually we see institutes offering objective based learning which usually causes a lag behind in subjective examinations which is the pattern followed by schools. I think it is really a work of planning to make us students grab the advantages of modes of examination, Objective Subjective and Onli...

Anika Saxena

8th

In terms of methodology I want to say that institute provides expert guidence and results oriented monitering supplements by requsite study material along with regular tests which help the students to improve their education skills.The techniques of providing education helps the students to asses...

Aman Kumar Shrivastava

10th

Abhyas academy is great place to learn. I have learnt a lot here they have finished my fear of not answering.It has created a habit of self studying in me.The teachers here are very supportive and helpful. Earlier my maths and science was good but now it has been much better than before.

Barkha Arora

10th

Abhyas institute is one of the best coaching institute in the vicinity of Ambala cantt.The institute provides good and quality education to the students.The teachers are well experienced and are v...

Abhyas institute is one of the best coaching institute in the vicinity of Ambala cantt.The institute provides good and quality education to the students.The teachers are well experienced and are very helpful in solving the problems. The major advantages of the institute is extra classes for weak...

Shreya Shrivastava

8th

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

My experience with Abhyas academy is very nice or it can be said wonderful. I have been studying here from seven class. I have been completing my journey of three years. I am tinking that I should ...

Hridey Preet

9th

Abhyas is good institution and a innovative institute also. It is a good platform of beginners.Due to Abhyas,he has got knoweledge about reasoning and confidence.My son has improved his vocabulary ...