Nested Square Root Sum (And Triangular Numbers)

Algebra Level 5

The value of k=12015Tk+18Tk\sum_{k=1}^{2015} \sqrt{\sqrt{T_k+\dfrac{1}{8}}-\sqrt{T_k}} where TkT_k is the kkth triangular number, can be expressed as pqr\sqrt{\sqrt{\dfrac{p}{q}}-\sqrt{r}} for positive integers p,q,rp,q,r and gcd(p,q)=1\gcd(p,q) = 1. Find the last three digits of p+q+rp+q+r.

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