KMO 2009 Problem 8

For a positive integer $n$ , define a function $f_n (x)$ at an interval $[ 0, n+1 ]$ as $$f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 .$$ Let $a_n$ be the minimum value of $f_n (x)$ . Find the value of $$\sum_{n=1}^{11} (-1)^{n+1} a_n .$$