Problem 1 [48pts] (1) [10pts] Let P(n) be the statement that 1+2 + +n -n(n +1)/2, for every positive integer n Answer the following (as part of a proof by (weak) mathematical induction) 1. [2pts] Define the statement P(1) 2. [2pts] Show that P(1) is True, completing the basis step 3. [4pts] Show that if P(k) is True then P(k+1) is also True for k2l, completing the induction step [2pts] Explain why the basis step and induction step together show that P(n) is True for all n 4. (2) [15pts] Using (weak) mathematical induction, prove that nn is an odd positive integer, where n is a positive integer. Show all steps involved in the proof. (3) [15pts] Using weak mathematical induction, prove that if Ai, A2, ..., An, and Bi, B2, , Bn are sets, and n is any positive integer, then Show all steps involved in the proof. (4) [8pts] Suppose that P(n) is a propositional function. For the following properties of P(n), determine for which positive integers n, P(n) must be True: 1. [2pts] P(1) is True; and for all positive integers n, if P(n) is True then 2. [2pts] P(1) and P(2) are True; and for all positive integers n, if P(n) and 3. [2pts] P(1) is True; and for all positive integers n, if P(n) is True then P(2n) 4. [2pts] P(1) is True; and for all positive integers n, if P(n) is True, then P(n+2) is True P(n+1) are True, then P(n+2) is True is True P(n+1) is True