Fastest method to define whether a number is a triangular number

Question

A triangular number is the sum of the n natural numbers from 1 to n. What is the fastest method to find whether a given positive integer number is a triangular one?

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Answer 1

Home work ?

Sum of numbers from 1 to N

1 + 2 + 3 + 4 + ... n-1 + n

if you add the first plus last, and then the second plus the second from last, then ...

= (1+n) + (2+n-1) + (3+n-2) + (4+n-3) + ... (n/2 + n/2+1)

= (n=1) + (n+1) + (n+1) + ... (n+1); .... n/2 times

= n(n+1)/2

This should get you started ...