1. [18 points, 6 points for each sub-problem] For each of the following sequences. write out the next two numbers in the sequence, and then write its recursive and closed formula as well. a. 1,6,11,16,21. ,_ Recursive Closed b. 1,3,7,13,21,__, Recursive Closed c. 3,6,12,24,48,__.__ Recursive Closed 2. [15 points] Evaluate the following summations given in big sigma notations: 3. [12 points] Express the foilowing summations using the big sigma notations: a. 2+3+4+5+6+7+8 b. 3+6+9+12+15 4. [18 points. 3 points each] Let A=da.b.c} B={a.b,d U={a.b.c.d.e,f, g,h} (universal set) Answer the following questions on sets: 5. [10 points, 5 points each] Interpret the following set builder notations, and find out all members of each set. a. {n:ne Z and n =k for some integer k E {2,3,4} } b. {2n +1:ne A} where A={1,2,3}. 6. [15 points, 5 points each] True or false questions on sets. 6. [15 points, 5 points each] True or false questions on sets. a. Is | A-BHA|-| B | true for any two sets, A and B ? Why ? Under what conditions on sets A and B, the above equation holds? b. Is the following statement true? Explain your answer. If A E P(B), then A c B, here P(B) refers to the power set of B. c. Draw Venn Diagram to check whether the following equations hold: An ( BUC) =(AnB)u(Anc).T. [12 points, 6 points each] Draw truth table for the following propositions: a. Na b. p=>(\"II) 8. [10 points] Draw truth tables for the following two pairs of propositions, and comment on whether they are equivalent to each other. Also explain the result using your intuition (one way to do this is to make up concrete examples, for example let p stands for "it rains", q stands for "it's cold", and interpret each of the propositions and see if they mean the same thing). a. (pvq)A 1pv.q b- -(p=>q)ApA-a