Given
n3 + 2n is divisible by 3 ∀ n ∈ N
Proof
We will prove the given statement by induction
STEP 1
n = 1
n3 + 2n = 13 + 2*1 = 3
3 is divisible by 3. Therefore, the statement is true for n = 1
STEP 2
Let the given statement be true for n = k
k3 + 2k = 3x
Now, we need to prove that if the statement is true for n = k then it is also true for n = k + 1
(k + 1)3 + 2(k + 1) = k3 + 1 + 3k2 + 3k + 2k + 2
= (k3 + 2k) + 1 + 3k2 + 3k + 2
= 3x + 3k2 + 3k + 3
= 3 * ( x + k2 + k + 1 )
Thus, we can say that (k + 1)3 + 2(k + 1) is divisible by 3 if k3 + 2k is divisible by 3.
Therefore, we can say that if the given statement is true for n = k, then it is also true for n = k + 1. Hence, by the principle of mathematical induction, the given statement is true ∀ n ∈ N.