Prove that n^3+2n is divisible by 3

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.