Consider the following inductive step of a proof by induction of the statement "n^3 + 2n is divisible by 3 for every positive integer n" by induction. Assume that k^3 + 2k is divisible by 3 for some k greaterthanorequalto 1. We want to show that (k + 1)^3 + 2(k + 1) is divisible by 3. We have that: (k + 1)^3 + 2(k + 1) = k^3 + 3k^2 + 3k + 1 + 2k + 2 = (k^3 + 2k) + (3k^2 + 3k + 3) (k^3 + 2k) is divisible by 3 because ___. (3k^2 + 3k + 3) is divisible by 3 because ___. Therefore, (k + 1)^3 + 2(k + 1) is divisible by 3. Statement (1) could be "of the inductive hypothesis", Statement (2) could be "all terms have common factor 3" Statement (1) could be "all terms have common factor 3", Statement (2) could be "of the inductive hypothesis" Statement (1) could be "of the inductive hypothesis", Statement (2) could be "of the inductive hypothesis" Statement (1) could be "all terms have common factor 3", Statement (2) could be "all terms have common factor 3" Recall that H_n is the n-th harmonic number: H_n = 1 + 1/2 + 1/3 + ... + 1. Suppose we are trying to prove that H_2^n lessthanorequalto 1 + n for all non-negative integers n by induction. In the inductive step, we must show that H_2^k + 1 lessthanorequalto 1 + (k + 1). Which of the following is a reasonable first step in this part of the proof? H_2^k + 1 = H_2^k + H_2^k + 1 H_2^k + 1 = H_2^k + H_2^k + 1 H_2^k + 1 = H_2^k + 1/2^k + 1 + 1/2^k + 2 + ... + 1/2^k + 2^k H_2^k + 1 = H_2^k + 1/2^k + 1 + 1/2^k + 2 + ... + 1/2^k + k Let A = {a, b, c, d, e, f, g, h} and R = {(a, a), (b, b), (c, d), (c, g), (d, p), (e, h), (f, f), (g, g)} be a relation on A. Then R is: reflexive and symmetric antisymmetric and not reflexive reflexive and antisymmetric symmetric and not reflexive