FI-MO 1999 Problem 2

Suppose that the positive numbers $a_1, a_2,.. , a_n$ form an arithmetic progression; hence $a_{k+1}- a_k = d,$ for $k = 1, 2,... , n - 1.$ Prove that $$\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.$$