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Expression for x_n in terms of n and x_1

Source: Chinese MO 2004

September 25, 2011
floor functionmodular arithmeticinductionalgebra proposedalgebra

Problem Statement

Let cc be a positive integer. Consider the sequence x1,x2,…x_1,x_2,\ldots which satisfies x1=cx_1=c and, for n≥2n\ge 2, xn=xn−1+⌊2xn−1−(n+2)n⌋+1x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1 where ⌊x⌋\lfloor x\rfloor denotes the largest integer not greater than xx. Determine an expression for xnx_n in terms of nn and cc.
Huang Yumin
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