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Show there exist a, b such that limit is 1

Source: Turkey IMO TST 1995 #6

July 8, 2011
limitalgebra unsolvedalgebra

Problem Statement

The sequence {xn}\{x_n\} of real numbers is defined by x_1=1  \text{and}  x_{n+1}=x_n+\sqrt[3]{x_n}  \text{for}  n\geq 1.
Show that there exist real numbers a,ba, b such that lim⁡n→∞xnanb=1\lim_{n \rightarrow \infty}\frac{x_n}{an^b} = 1.
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