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Let a sequence Xo, X1,X2,... be defined in the following way: X12 1) Compute the first 10 terms of this sequence. (2 points) 2) Prove that this sequence is strictly increasing, .e., Vn 20:X >X. (2 points) 3) Prove that Vn 20: Xn S4. What are the base cases? What is the inductive step? (5 points) 4) The above result suggests that this sequence grows in the worst case exponentially, i.e., X 0(4). Consider trying to tighten this bound in the following way: what are the smallest values a and B such that the above proof still works for Vn 20:XnS? Give the tightest bounds on Xn you can. (10 points) 5) What is the tightest lower bound on Xn you can achieve in this way? Characterize the long term /asymptotic behavior of X, as best you can. (5 points)

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