By induction prove that Vn E Z+, Problem 2. [25 points] By induction prove that Vn...

 

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By induction prove that Vn E Z+, Problem 2. [25 points] By induction prove that Vn € Z+, IM³ 2 n(n + 1)(2n+1) 6 (3)-3 m Hint: recall from class that Σ i = m(m+1) 2 i=1 Problem 3. [25 points] By induction prove that Vn € Z+, n 2² (i + 1) = = n. 2n+1 i=1 Problem 4. [25 points] a) [12 points] Prove by induction: for all n = N, 2 (n² + n) Prove by induction: for all n Є N, b) [13 points] 6 | (n³ - n) Hint: you might want to solve the following equation for x: (k+1)³ − (k + 1) = (k³ − k) + x ne rs By induction prove that Vn E Z+, Problem 2. [25 points] By induction prove that Vn € Z+, IM³ 2 n(n + 1)(2n+1) 6 (3)-3 m Hint: recall from class that Σ i = m(m+1) 2 i=1 Problem 3. [25 points] By induction prove that Vn € Z+, n 2² (i + 1) = = n. 2n+1 i=1 Problem 4. [25 points] a) [12 points] Prove by induction: for all n = N, 2 (n² + n) Prove by induction: for all n Є N, b) [13 points] 6 | (n³ - n) Hint: you might want to solve the following equation for x: (k+1)³ − (k + 1) = (k³ − k) + x ne rs