Interesting problem

Algebra Level 4

\[\sqrt{1+\frac 1{1^2} + \frac 1{2^2}} + \sqrt{1+\frac 1{2^2} + \frac 1{3^2}} + \sqrt{1+\frac 1{3^2} + \frac 1{4^2}} + \cdots + \sqrt{1+\frac 1{2014^2} + \frac 1{2015^2}} \]

If the sum above can be expressed as \(\dfrac {ab}c\), where \(a\), \(b\) and \(c\) are positive integers with \(ab\) and \(c\) being coprime integers and \(a = b+2\). Find \(a+b+c\).