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 $a+b+c$ .