I need help with e, f.

Exercise 4.12.2: Proving identities by induction. i About n2 Prove each of the following statements using mathematical induction (a) 2 n(n+1) Prove that for any positive integer n, n 2 j=1 (b) n Prove that for any positive integer n. j. 21 j. 21 = (n 1)2n+1 +2 j=1 (C) n Prove that for any positive integer n, i j - 1) n(n2 1) 3 j=1 (d) n 1 1 1 Prove that for any positive integer n, = 1 (j +1) n+1 j=1 (e) Prove that for any non-negative integer n, n Lj. 3 3 n.3n+1 4 (n + 1)3" + 1] j=0 (f) Prove that for any positive integer n 2, 1 1 1 1 1 1 22 )... n +1 2n 32 m2