Use the principle of mathematical induction to prove that 2 | (n^2 + n) for all n greaterthanorequalto 0. Use the principle of mathematical induction to prove that 2n + 3 lessthanorequalto 2^n for all n greaterthanorequalto 4. Give a recursive definition of: The set of positive and negative even integers The set of negative integer powers of 2: {1, 1/2, 1/4, 1/8, middot middot middot} Give a recursive definition of: The set of strings {1, 11, 111, 1111, 11111, ...} The function f (n) = n + 1/3, n = 1, 2, 3, ... Use a merge sort to sort 4, 2, 3, 9, 5, 8, 7, 6 into increasing order. Show all the steps used by the algorithm. Give a recursive algorithm for computing na where n is a positive integer and a is a real number