Algebra Level 4

$\large f(x) = \dfrac{(\sqrt{x} + 1)(x^{2} - \sqrt{x})}{x \sqrt{x} + x + \sqrt{x}}$

Define $$f(x)$$ as above and further define $$g(x) = f(f(x))$$.

Find the value of $$\large\displaystyle [\sum_{i=2}^{1000} g(i)] - 1$$.

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