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Integer part of a sequence

Source: Finnish Mathematics Competition 2003, Final Round, Problem 2

November 14, 2011
algebra unsolvedalgebra

Problem Statement

Find consecutive integers bounding the expression \frac{1}{x_1 + 1}+\frac{1}{x_2 + 1}+\frac{1}{x_3 + 1}+... +\frac{1}{x_{2001} + 1}+\frac{1}{x_{2002} + 1} where x1=1/3x_1 = 1/3 and xn+1=xn2+xn.x_{n+1} = x_n^2 + x_n.
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