Olympiad Problem 4

Algebra Level 5

The sequence xn{x}_{n} is defined by x1=12,   xk+1=xk2+xk.x_1=\dfrac{1}{2},~~~ x_{k+1} = x_{k}^2 + x_k. If S=1x1+1+1x2+1+1x3+1++1x100+1S=\dfrac{1}{{x}_{1}+1}+\dfrac{1}{{x}_{2}+1} + \dfrac{1}{{x}_{3}+1} +\cdots + \dfrac{1}{{x}_{100}+1} then find the value of S\left\lfloor S\right\rfloor.

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