Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer...

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Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, Σs = - (n(n + ¹) ) ² 2 j=1 (b) Prove that for any positive integer n, Σj.2³=(n-1)2+¹ +2 j=1 (c) Prove that for any positive integer n, Σj(j-1) = n(n²-1) 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 Σj· 3³ = n · 3n+¹ −(n+1)3″ . j=0 (f) Prove that for any positive integer n ≥ 2, 2/²) (1 - 31/7)... (1₁ - 12/17) 3² n² Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, Σs = - (n(n + ¹) ) ² 2 j=1 (b) Prove that for any positive integer n, Σj.2³=(n-1)2+¹ +2 j=1 (c) Prove that for any positive integer n, Σj(j-1) = n(n²-1) 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 Σj· 3³ = n · 3n+¹ −(n+1)3″ . j=0 (f) Prove that for any positive integer n ≥ 2, 2/²) (1 - 31/7)... (1₁ - 12/17) 3² n²