1 1. For a proof by mathematical induction of For every natural number n 6, 2n > n2 + 1 the base case should be n = 10 points Question 2 1. A proof by induction of For every natural number n, n7 + 6n is a multiple of 7 could begin with o Base case: For n = 0, 07 + 6 * 0 = 0 which is a multiple of 7 What is the appropriate next step? Suppose that n8 + 7n is a multiple of 8. Let n be an arbitrary natural number and suppose that (n+1)7 + 6(n+1) is a multiple of 7. Let n be an arbitrary natural number and suppose that n7 + 6n is a multiple of 7. Consider (n+1)7 + 6(n+1) 10 points Question 3 1. Place the following 4 main steps of an induction proof into an appropriate order. - State the result that you intend to prove. Supposing that this result is true for n, prove that this result is true for n + 1. Summarize what you have proven. Prove that this result is true for a base case, such as for n = 0 or for n = 1. 10 points Question 4 1. Place the following 5 steps into an appropriate order for a proof by mathematical induction of For every natural number n 1, 12 * 1 + 12 * 2 + ... 12 * n = 6 n (n+1) - Verify that 12 * 1 + 12 * 2 + ... 12 * (n+1) = 6 (n+1) ((n+1) + 1) Suppose that 12 * 1 + 12 * 2 + ... 12 * n = 6 n (n+1) Verify that 12 * 1 + 12 * 2 + ... 12 * n = 6 n (n+1) in this case Consider the base case of n = 1 Let n be an arbitrary natural number 10 points Question 5 1. The following is an incorrect proof of For every natural number n 1, in every set of n cars, all the cars are blue Proof by induction: o Induction step: Let n be an arbitrary natural number and suppose that in every set of n cars, all the cars are blue. Consider a set of n+1 cars. Label the n+1 cars as car1, car2, ..., carn+1 o Consider car2, ..., carn+1, this is a set of n cars. By the induction assumption, all those cars are blue. In particular, carn+1 is blue and thus all n+1 cars are blue. This induction proof is obviously incorrect. What's wrong with it? We didn't verify the base n = 1 case We didn't verify the base n = 0 case The proof of the "true for n" step implying "true for n+1" step is not valid when n = 1 The statement should be that all cars are silver 5 points