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Find limit

Source: Vietnam TST 1990, Problem 2

July 29, 2008
limitalgebra unsolvedalgebra

Problem Statement

Let be given four positive real numbers a a, b b, A A, B B. Consider a sequence of real numbers x1 x_1, x2 x_2, x3 x_3, … \ldots is given by x_1 \equal{} a, x_2 \equal{} b and x_{n \plus{} 1} \equal{} A\sqrt [3]{x_n^2} \plus{} B\sqrt [3]{x_{n \minus{} 1}^2} ( n \equal{} 2, 3, 4, \ldots). Prove that there exist limit \lim_{n\to \plus{} \propto}x_n and find this limit.
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