AUSTRIAN-POLISH 1988 Problem 2

If $a_1 \le a_2 \le .. \le a_n$ are natural numbers ( $n \ge 2$ ), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0$$ holds for all $n$ -tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$ .