ILL 1985 Problem 13

Find the average of the quantity $$(a_1 - a_2)^2 + (a_2 - a_3)^2 +\cdots + (a_{n-1} -a_n)^2$$ taken over all permutations $(a_1, a_2, \dots , a_n)$ of $(1, 2, \dots , n).$