Please complete all parts of the question as they are connected (parts a, b, c, d, e). Please show all work. Thank you.

Problem 2: Proofs by induction 13 +23 +33 + . . . + n-n2(n + 1)2 1-2 2-3 3-4+1 n(n+1)n+1 1-20 +2-21+3-22+..n 2-1(n1)2" +1 (d) n2 1 is a multiple of 4 if n is odd. Hint: your inductive step for n should consider n - 2 and not n- 1. (e) Show that every positive integer can be written as the product of an odd number and a power of 2