2. Prove or disprove each of the following conjectures, using one of the techniques explained in Chapter 2.1. State the name of the technique that you used. Remember that you can use facts about integers, such as the fact that an even number can be expressed as 2k, where k is an integer, and an odd number can be expressed as 2k +1. a. The quotient of two even integers is an even integer. b. The sum of any two consecutive integers is odd. Consecutive integers occur as pairs, e.g., 6 & 7, 101 & 102,-47 & -46. C. If x and y are odd numbers then x-y is an odd number. d. The sum of an integer and its square is even. 3. Use inductive reasoning to develop a conjecture for each of the following. Be sure to show your reasoning, a. The sum of any three consecutive integers is/is not even. b. The difference between two even integers is/is not odd. 4. State the contrapositive of each of the following statements. Hint: First identify the antecedent and the consequent. a. Water and sun are necessary conditions for healthy plants. b. If a person studies hard she will make good grades. C. I cannot get to work if my car is broken END OF HWK 2 5. Use mathematical induction to prove that the following statements are true for every positive integer n. a. 2 + 4 + 6 + ... + 2n = n(n+1) b. 103+2 4+3 5+ ... + n(n + 2) = (n(n + 1)(2n +7))/ 6 the symbol (represents multiplication) 6. Write the first 5 values in the following sequence: D(1) = 3 D(2) = 5 D(n) = n n - D(n - 1) + (n-1) - Dn - 2) 7. Let az, az, az...., and b, b2b3 both be sequences that satisfy the recurrence relation that the nth term equals 3 times the (n-1)st term for all integers n 1: an = 3 an-1, bn = 3bn-1 But suppose the initial conditions for the sequences are different: a1 = 0, b1 = 1. Find a. az, a3, 24, b. b2, bz, b4