Use mathematical induction to prove the following claims: (a) For all natural numbers n ≥ 2, 3n > n². (b) For all natural numbers n ≥ 7, 3" <n!. (Feel free to use a calculator for the base step.) Hint: (n+1)! (n + 1) ⋅ n! = n (c) For all natural numbers n ≥ 1, 2i = n(n + 1). i=1 ጥሪ (d) For all natural numbers n ≥ 1, 2-1 = 21. n i=1 - (e) For all nЄ N, Σ i! · i = (n + 1)! — 1. i=0 (f) For all nЄN, n² 3n is even. (Note: It is possible to prove this without induc- tion. In fact, it might even be easier to prove it without induction. But for this assignment, you must provide an induction proof.)