The term in the interval (n(n-1)/2, n(n+1)/2]is n. The pattern generalizes as follows: The first term is 1, the next two terms is 2, etc... (as you already know). So, every number (k) repeats k times, and the successive terms increase by 1, after repetition, starting from 1. It means, the nth term has to do with how many numbers are below it, and since each number repeats itself "its value" number of times, this has to do with sum of first n numbers. Once you cross, sum of first n numbers terms, you get a new value for the term in the series.

Salahuddin

Maths online