Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, s = - (n(n + ) ) 2 j=1 (b) Prove that for any positive integer n, j.2=(n-1)2+ +2 j=1 (c) Prove that for any positive integer n, j(j-1) = n(n-1) 3 j=1 (d) Prove that for any positive integer n, 72 1 j(j+1) - 1 n+1 (e) Prove that for any non-negative integer n, 3 72 j 3 = n 3n+ (n+1)3 . j=0 (f) Prove that for any positive integer n 2, 2/) (1 - 31/7)... (1 - 12/17) 3 n