A triangular number is defined as a number which has the property of being expressed as a sum of consecutive numbers starting with 1. How many triangular numbers less than 1000, have the property that they are the difference of squares of two consecutive natural numbers?

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Please find below the solution to the asked query:

We know " This sequence comes from a pattern of dots that form a triangle "

Triangular numbers :  1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666 ...

So, n^th triangular number  = $\frac{\mathrm{n} \left(\mathrm{n} + 1 \right)}{2}$  , So 44^th triangular number  = $\frac{44 \left( 44+ 1 \right)}{2} = 22 \times 45$   = 990 after that we get triangular number greater than 1000 .  and we can see that there are same number of odd and even numbers in triangular number , So in 44 triangular number 22 odd number and 22 even numbers are there .

And we know difference of squares of two consecutive natural numbers : 3 , 5 , 7 , ...  all odd number

So, from above condition we can say there are 21 triangular numbers that can satisfied the given condition ( Here we neglect number = 1 as that is not a difference of consecutive natural number ) .                            ( Ans )

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