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find real numbers s.t. sequence has finite limit

Source: Vietnam TST 1993 for the 34nd IMO, problem 2

June 25, 2005
algebra unsolvedalgebra

Problem Statement

A sequence {an}\{a_n\} is defined by: a1=1,an+1=an+1ana_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}} for n=1,2,3,…n = 1, 2, 3, \ldots. Find all real numbers qq such that the sequence {un}\{u_n\} defined by un=anqu_n = a_n^q, n=1,2,3,…n = 1, 2, 3, \ldots has nonzero finite limit when nn goes to infinity. THERE MIGHT BE A TYPO!
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