VN-TST 1993 Problem 2

A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$ . Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$ , $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity.

THERE MIGHT BE A TYPO!