Sum of roots with a twist

Calculus Level 3

Let \(\alpha_{n}, \beta_{n}\) be the distinct roots of the equation \(x^{2}+(n+1)x+n^{2} = 0\). If

\[\sum_{n=2}^{2015} \frac{1}{(\alpha_{n}+1)(\beta_{n}+1)}\]

can be expressed in the form \(\frac{a}{b}\), where \(a\) and \(b\) are positive coprime integers, find \(b-a\).

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