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Algebra Level 4

f(x)=(x+1)(x2x)xx+x+x\large f(x) = \dfrac{(\sqrt{x} + 1)(x^{2} - \sqrt{x})}{x \sqrt{x} + x + \sqrt{x}}

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

Find the value of [i=21000g(i)]1\large\displaystyle [\sum_{i=2}^{1000} g(i)] - 1.

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