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x_n = \pm (n−1)x_{n−1} \pm (n−2)x_{n−2} \pm ... \pm 2x_2 \pm x_1

Source: Czech and Slovak MO, III A, 2003 p3

January 12, 2020
recurrence relationRecurrencecombinatoricsalgebra

Problem Statement

A sequence (xn)n=1∞(x_n)_{n= 1}^{\infty} satisfies x1=1x_1 = 1 and for each n>1,xn=±(n−1)xn−1±(n−2)xn−2±...±2x2±x1n > 1, x_n = \pm (n-1)x_{n-1} \pm (n-2)x_{n-2} \pm ... \pm 2x_2 \pm x_1. Prove that the signs ” ±\pm” can be chosen so that xn≠12x_n \ne 12 holds only for finitely many nn.
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