$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+...+\frac{1}{999\cdot 1000}+1$
= $\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{999}-\frac{1}{1000}+1$
= $\frac{1}{1}-\frac{1}{1000}+1$
=$\frac{999}{1000}+1$
=$\frac{999}{1000}+\frac{1000}{1000}$
=$\frac{1999}{1000}$
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