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PROBLEM 4 (10 points) Suppose T(0) = 0 and for all n > 0, T(n+1) = rT(n) + a, where r and a are constants. Prove by induction that for all n > 0: I(n)=al-nn = 1-r Solution. PROBLEM 5 (5+5 points) (A) How many different terms of the form xxxxx where i > 3, j 2, k 1, are equal to x13? (B) Arrange the following in Big-O order, so if f comes before g, then f is O(g). 4n², log3(n), log2(n!), nn, :1. 3n, n log2(n), 2n+1, 17, n!, 22n Solution. PROBLEM 6 (5 points) Does this series converge or diverge? Explain why. Solution. PROBLEM 7 (2+2+2+2 points) Determine which of the following properties are equivalence relations. If the relation is an equiv- alence relation, describe the equivalence classes. Otherwise, explain why it is not an equivalence relation. (A) The relation R on Z where aRb iff a < b (B) The relation R on pairs of real numbers where iff y = y2. For each of the following, state whether the set is finite, countably infinite, or uncountably infinite. (C) The set of numbers of the form a + bV2 where a, b € N. (D) The powerset of the set of all sentences that contain fewer than 10 words. (A word is something listed in some standard dictionary.) Solution.

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