Suppose s is the set of numbers recursively defined by: Use structural induction to prove that all members of S are positive in tegers with a last decimal digit 7. Start by using the division algorithm to define rigorously what it means for a positive integer to have a last decimal digit 7. Find a positive integer n that has a last decimal digit 7 and is not in the set S from the previous problem. Prove that n is not in S Formulate a principle of induction that is suitable for proving a closed form an-f(n) for a sequence defined by two initial values a2 and a3, and a two-step recurrence an+2 g(an+1,an) for n 2 2. You do not have to prove the correctness of this principle. Apply it to prove that your closed form from the previous part is correct.