Use mathematical induction to prove that the following

Question

Use mathematical induction to prove that the following statements are true.

1. For n ≥ 1, the sum of the first n positive integers equals n(n+1)/2 , i.e.,

       1 + 2 + 3 + . . . + n = n(n + 1)/2.

2. For n ≥ 1, the sum of the first n positive square numbers equals n(n+1)(2n+1)/6 , i.e.,

      1² + 2² + 3² + . . . + n² = n(n + 1)(2n + 1)/6.

3. For n ≥ 1, the sum of the first n positive cube numbers equals the square of the sum of the first n positive integers, i.e.,

     1³ + 2³ + 3³ + . . . + n³ = [1 + 2 + 3 + . . . + n]².

4. For n ≥ 1, the number n² - n is divisible by 2.

5. For n ≥ 1, the number n³ - n is divisible by 3.

6. For n ≥ 4, 2ⁿ < n!.