Periodicities which preserve and periodicities which destroy boundedness

It is known that every positive solution of the difference equation with positive parameter β > 0 is bounded. In this note we study the difference equation . We show that every positive solution of this equation is bounded when { β n } n =0 ∞ is a period-2 sequence of positive real numbers, that is, “Period-2 Preserves Boundedness.” We also show that there exist prime period-3 m , sequences { β n } n =0 ∞ of positive real numbers such that the equation has unbounded solutions. That is,“Period-3 m Destroys Boundedness.”