ulation ▁ ▁The ▁Ak ra – B az zi ▁method ▁applies ▁to ▁rec urrence ▁formulas ▁of ▁the ▁form ▁ ▁The ▁conditions ▁for ▁usage ▁are : ▁▁ ▁sufficient ▁base ▁cases ▁are ▁provided ▁▁ ▁and ▁ ▁are ▁constants ▁for ▁all ▁▁▁ ▁for ▁all ▁▁▁ ▁for ▁all ▁▁ ▁, ▁where ▁c ▁is ▁a ▁constant ▁and ▁O ▁not ates ▁Big ▁O ▁notation ▁▁ ▁for ▁all ▁▁▁ ▁is ▁a ▁constant ▁ ▁The ▁asympt otic ▁behavior ▁of ▁ ▁is ▁found ▁by ▁determ ining ▁the ▁value ▁of ▁ ▁for ▁which ▁ ▁and ▁plug ging ▁that ▁value ▁into ▁the ▁equation ▁ ▁( see ▁ Θ ). ▁Int uit ively , ▁ ▁represents ▁a ▁small ▁perturb ation ▁in ▁the ▁index ▁of ▁. ▁ ▁By ▁not ing ▁that ▁ ▁and ▁that ▁the ▁absolute ▁value ▁of ▁ ▁is ▁always ▁between ▁ 0 ▁and ▁ 1 , ▁ ▁can ▁be ▁used ▁to ▁ignore ▁the ▁floor ▁function ▁in ▁the ▁index . ▁ ▁Similarly , ▁one ▁can ▁also ▁ignore ▁the ▁ce iling ▁function . ▁ ▁For ▁example , ▁ ▁and ▁ ▁will , ▁as ▁per ▁the ▁Ak ra – B az zi ▁theorem , ▁have ▁the ▁same ▁asympt otic ▁behavior . ▁ ▁Example ▁ ▁Suppose ▁ ▁is ▁defined ▁as ▁ 1 ▁for ▁integers ▁ ▁and ▁ ▁for ▁integers ▁. ▁ ▁In ▁applying ▁the ▁Ak ra – B az zi ▁method , ▁the ▁first ▁step ▁is ▁to ▁find ▁the ▁value ▁of ▁ ▁for ▁which ▁. ▁ ▁In ▁this ▁example , ▁. ▁ ▁Then , ▁using ▁the ▁formula , ▁the ▁asympt otic ▁behavior ▁can ▁be ▁determined ▁as ▁follows :