Reference no: EM13975742
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!.
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