Suppose s is the set of numbers recursively defined

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4. SupposeSis the set of numbers recursively defined by:1SxS3xSxSx2S.Use structural induction to prove that all members ofSare powers of 3.Here, by a power of 3, we mean a number of the form 3kwherekis anon-negative integer. Do not confuse ”power of 3” with ”multiple of 3”and make sure to review the laws of exponentiation for this problem.5. A sequence is defined bya0= 0 anda1= 1, andan= 3an-1-2an-2foralln2. Use the method covered in the lecture to find a closed form foranand show your work. Start by writing the characteristic equation.Set B:1. Use induction to prove the inequalitynXk=11k2n-1.for all positive integersn.2. SupposeBis the set of bit strings recursively defined by:001SbS11bSbS10bSbS0bS.Letanthe number of bit strings inBof lengthn, forn2. Determine arecursive definition foran, i.e. determinea2,a3and a recurrence relation.Make sure to justify your recurrence relation carefully. In particular, youmust make it clear that you are not double-counting bit strings.3. Find a closed form solution for the sequenceanfrom the previous prob-lem using the method for solving linear, homogeneous, constant-coefficient

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