Can it be summed up?

Algebra Level 4

\[ \large \sum_{n = 1}^{9800} \dfrac{1}{\sqrt{n + \sqrt{n^2 - 1}}} \]

If the sum above can be expressed as \(p + q \sqrt{r}\), where \(p, q\) and \(r\) are positive integers and \(r\) is not divisible by the square of any prime, determine \(p + q + r\).


This problem is adopted form a mock test of a renowned contest.
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