OMCC 2015 Problem 2

A sequence $(a_n)$ of real numbers is defined by $a_0=1$ , $a_1=2015$ and for all $n\geq1$ , we have $$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$ Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$ .