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Recurrence relation, maximum of x1

Source: Mongolia MO 2001 Grade 10 P1

April 12, 2021
Sequencerecurrence relationalgebra

Problem Statement

Suppose that a sequence x1,x2,…,x2001x_1,x_2,\ldots,x_{2001} of positive real numbers satisfies 3xn+12=7xnxn+1−3xn+1−2xn2+xn and x37=x2001.3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.Find the maximum possible value of x1x_1.
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