Question

how to draw a a consecutive triangular no. as it very is difficult to solve questions like:
how can the number 36 be expressed as the sum of two consecutive square numbers?

Answer 1

Answer ;

We know triangular numbers are : 1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , 45 , ...

We can represent n^th triangular number = $\frac{n\left(n+1\right)}{2}$

So , From this formual , we can find any two consecutive triangular number ,

As : we want to find 15^th and 16^th triangular number , So

15^th triangular number = $\frac{15\left(15+1\right)}{2}=\frac{15\times 16}{2}=15\times 8=120$
And
16^th triangular number = $\frac{16\left(16+1\right)}{2}=\frac{16\times 17}{2}=8\times 17=136$

So , Two consecutive triangular numbers ( 15^th and 16^th ) = 120 and 136

And

We represent square number as : 0² = 0 , 1² = 1 , 2² = 4 , 3² = 9 , ...

So,
Let two consecutive square numbers : x² and ( x + 1 )²
So we can represent
The number 36 be expressed as the sum of two consecutive square numbers.

x² + ( x + 1 )² = 36 ( Ans )