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 nth 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
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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

find (99)^{2}-(96)^{2}

Right Option : B
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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

and find (111)^{2}-(109)^{2}

Right Option : D
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Explanation
Question : 3

Express (21)^{2} as the sum of two consecutive natural numbers.

Right Option : A
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Explanation
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