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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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