PLEASE ANSWER CLEARLY AND CORRECTLY ALL THE QUESTIONS BELOW

1. Which type of proof is being used in each case? (UODpositive integers) Prove: If n is odd then (n+1 )2 is even (a) 1. n is odd 2. (n+1) is odd 3. some inference... (b) 1. (n+1) is odd 2. some inference.... (c) 1. n is odd 2. some inference.. 2. Write a proof by contradiction of this proposition: If m and n are perfect squares, then m'n is a perfect square 3. Write a contrapositive proof of this proposition: If m is a positive integer, then mn 4. For an inductive proof of the proposition P(n): 1*1!+ 2"2! +... + (n-1)(n-1) n!-1 for all positive integers, n > 1 (a) Write the basis step (hint:start at 2, since n > 1) (b) Write the Inductive Hypothesis (c) Complete the proof 5. For an inductive proof of the P(n): nn n! for all positive integers, n (a) Write the basis step (b) Write the Inductive Hypothesis (c) Complete the proof