Find a triple $(l, m, n)$ of positive integers $(1<l<m<n)$, such that $\sum_{k=1}^{l}k, \sum_{k=l+1}^{m}k, \sum_{k=m+1}^{n}k$ form a geometric sequence in order.
Problem
Source: China south east mathematical Olympiad 2012 day1 problem 1
Tags: geometric sequence, algebra unsolved, algebra