Use mathematical induction to prove the statement is true for all positive integers n.3 is a factor of n3 + 2n

Question

What will be an ideal response?

Answer 1

Answers may vary. Possible answer:
First, we show the statement is true when n = 1.
For n = 1, 3 is a factor of 13 + 2 ? 1 = 3
So P1 is true and the first condition for the principle of induction is satisfied.
Next, we assume the statement holds for some unspecified natural number k. That is,
Pk: 3 is a factor of k3 + 2k is assumed true.
On the basis of the assumption that Pk is true, we need to show that Pk+1 is true. 
Pk+1: 3 is a factor of (k + 1)3 + 2(k + 1).
(k + 1)3 + 2(k + 1) = (k3 + 3k2 + 3k + 1) + (2k + 2)
(k + 1)3 + 2(k + 1) = k3 + 2k + 3k2 + 3k + 3
(k + 1)3 + 2(k + 1) = (k3 + 2k) + 3(k2 + k + 1)
Since Pk is assumed true, 3 is a factor of (k3 + 2k) . Also 3 is a factor of 3(k2 + k + 1). So 3 is a factor of (k + 1)3 + 2(k + 1). 
So Pk+1 is true if Pk is assumed true. Therefore, by the principle of mathematical induction, 3 is a factor of n3 + 2n for all natural numbers n.