# 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?

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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