TR-MO 1999 Problem 4

Find all sequences ${{a}_{1}},{{a}_{2}},...,{{a}_{2000}}$ of real numbers such that $\sum\limits_{n=1}^{2000}{{{a}_{n}}=1999}$ and such that $\frac{1}{2}<{{a}_{n}}<1$ and ${{a}_{n+1}}={{a}_{n}}(2-{{a}_{n}})$ for all $n\ge 1$ .