Question 2 Sequences with recursive formulae (10 points) Describe each sequence in the following recursively. Please write initial conditions and assume that the sequences begin with a1. Note that, the initial condition of a sequence refers to the terms by which the sequence can be generated. Thus, the initial condition of a sequence is not necessary to only have one term. 1. on = 5" 2. The Fibonacci numbers 3. 1, 111, 11111, 1111111, ... 4. 12, 22, 3-, 4', ... 5. on = the number of subsets of a set of size n Question 3 Geometric progression (10 points) Suppose that inflation continues at three percent annually. (That is, an item that costs $1.00 now will cost $1.03 next year.) Let an = the value (that is, the purchasing power) of one dollar after n years. 1. Find a recurrence relation for an. 2. What is the value of $1.00 after 20 years? 3. What is the value of $1.00 after 80 years? 4. If inflation were to continue at ten percent annually, find the value of $1.00 after 20 years. 5. If inflation were to continue at ten percent annually, find the value of $1.00 after 80 years. Question 4 Summations (10 points) 1. Find the sum 112 + 113 + 114 + . .. + 673 . 2. Find the sum 2 - 4+ 8 - 16 + 32 -... - 228. 3. Find > ((-2)' - 2"). 4. Find in (2 x 3) + 3 x 2). 5. Find En Eryj.Question 5 Set cardinality (20 points) Determine whether each of the following sets is finite, countably infinite, or uncountable. For those that are finite, please give their cardinalities. For those that are countably infinite, please present a one-to-one correspondence between the set of positive integers and that set. 1. the integers with absolute value less than 1, 000, 000 2. the real numbers between 0 and 2 3. the set A x Z* where A = {2, 3) and we denote the positive integer set as Z+ 4. the integers that are multiples of 10 Question 6 Infinite sets and their cardinalities (20 points) 1. Show that (0, 1) has the same cardinality as [0, 2). 2. Show that (0, 1] and R have the same cardinality, where R denotes the real number set. Question 7 Matrices (20 points) 1017 010 Suppose A = 011 and B = 011 110 100 1. Find the join of A and B. 2. Find the meet of A and B. 3. Find A * B * A. 4. Find B * A * B. (We define the joint of A and B is the zero-one matrix with (i, j) th entry ay Vby, which is denoted as A V B; the meet of A and B is the zero-one matrix with (i, j)th entry ay Aby, which is denoted as AAB)