Once floored, always floored

Calculus Level 5

\[\large S_n(x) = \sum_{r=1}^{n} \lfloor rx \rfloor \]

For \( x \) \( \in \mathbb{R} \), define \(S_n(x) \) as above.

Evaluate

\[\displaystyle \bigg\lfloor \lim_{n \to \infty } \frac{S_n(x)}{n^2} \bigg|_{x = 2015.20162017} \bigg\rfloor \]

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