Relationship for Triangular numbers and Square numbers

The square number 25 as sum of the triangular numbers 10 and 15

The square number 36 as sum of the triangular numbers 15 and 21

The square number 49 as sum of the triangular numbers 21 and 28

  n th square is    nxn  as sum of the    n th triangular number  and  ( n-1) th th triangular number

Triangular numbers

Square numbers

Square numbers   

Square numbers

When a natural number is multiplied by itself number obtained as the answer is a square numbers.

       




                      1 x 1 =1        2 x 2 = 4       3 x 3 = 9     4 x 4 = 16    5 x 5 = 25

1st     => 1 x 1 = 1       =  1²                                                  2nd    => 2 x 2 = 4       =  2²                               

3rd     =>3 x 3 = 9        =  3² 

4th     => 4 x 4 = 16      =   4² 

5th     => 5 x 5 = 25      =   5²

n      => n x n  =          =    n²

Odd numbers and Even numbers.

Odd numbers

Any integer that can not be divided exactly by 2.

" (2n-1) "

n=1 => 2*1-1=1

n=2 => 2*2-1=3

n=3 => 2*3-1=5

(1,3,5,7,9......................2n-1)

Even numbers.

Even numbers can be divided evenly into groups of 2

"2n"

n=1 => 2*1 =2

n=2 => 2*2 =4

n=3 => 2*3 =6

(2,4,6,8,10........................2n)