CGMO 2003 Problem 6

Let $n \geq 2$ be an integer. Find the largest real number $\lambda$ such that the inequality $$a^2_n \geq \lambda \sum^{n-1}_{i=1} a_i + 2 \cdot a_n.$$ holds for any positive integers $a_1, a_2, \ldots a_n$ satisfying $a_1 < a_2 < \ldots < a_n.$