Nested Square Root Sum (And Triangular Numbers)

Algebra Level 5

The value of \[\sum_{k=1}^{2015} \sqrt{\sqrt{T_k+\dfrac{1}{8}}-\sqrt{T_k}} \] where \(T_k\) is the \(k\)th triangular number, can be expressed as \[\sqrt{\sqrt{\dfrac{p}{q}}-\sqrt{r}}\] for positive integers \(p,q,r\) and \(\gcd(p,q) = 1\). Find the last three digits of \(p+q+r\).

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