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Some more Interesting Patterns

1) Triangular Numbers :-
 


 

Triangular numbers are those numbers whose dot patterns can be arranged as triangles. Let us see what happens if we add two consecutive triangular numbers.
1 + 3 = 4 = 22

3 + 6 = 9 = 32

6 + 10 = 16 = 42

We observe that sum of two consecutive triangular numbers is a square number.


2. Adding Odd Numbers :-

The squares of a natural number n are equal to the sum of the first n odd numbers. Thus
12 = 1
22 = 1 + 3 : Sum of first 2 odd numbers
32 = 1 + 3 + 5 : Sum of first 3 odd numbers
82 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 : Sum of first 8 odd numbers.

In general we can write,

n2 = Sum of first n odd numbers


3. Difference of the squares of two consecutive numbers :-

The difference of the squares of two consecutive numbers is equal to the sum of the numbers.
(n + 1)2 - n2 = n2 + 2n + 1 = (n + 1) + n
Thus 112 - 102 = 121 - 100 = 21 = 11 + 10
 162 - 152 = 256 - 225 = 31 = 16 + 15


4. Easy method to find the square :-

(i) 672 = 4489

(ii) 6672 = 444889

(iii) 66672 = 44448889

Observe the following pattern, find the squares of 66667 and 666667.
 


 

5. Patterns for squares of 11, 111, 1111,:-
 

Proceeding like this we can get the squares of 11111112, 111111112 , ….





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