Recurring Styles 8 - Floor Function!

Algebra Level 5

$\large{S= \dfrac{x_1}{x_2} + \dfrac{x_2}{x_3} + \ldots + \dfrac{x_{2014}}{x_{2015}} + \dfrac{x_{2015}}{x_{2016}} }$

Define a sequence $$\large{(x_n)_{n \geq 1} }$$ by $$x_1 = \dfrac{1}{2015}$$ and $$x_{n+1} = x_n + x_n^2$$. Determine the value of $$\lfloor S \rfloor$$.

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