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x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}} cannot an be infitine sequence

Source: 2009 Belarus TST 2.2

November 8, 2020
algebraSequence

Problem Statement

a) Prove that there is not an infinte sequence (xn)(x_n), n=1,2,...n=1,2,... of positive real numbers satisfying the relation xn+2=xn+1−xnx_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}, ∀n∈N\forall n \in N (*) b) Do there exist sequences satisfying (*) and containing arbitrary many terms?
I.Voronovich
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