n³+2n is divisible by 3 for any natural number n

Question

The mathematical induction

Answer 1

1³ + 2(1) = 3

Assume

k³ + 2k = 3n

Demonstrate

(k + 1)³ + 2(k + 1) = 3m

k³ + 3k² + 3k + 1 + 2k + 2 = 3m

k³ + 2k + 3k² + 3k + 3 = 3m

3n + 3(k² + k + 1) = 3m

3(k² + k + 1) = 3m - 3n

3(k² + k + 1) = 3(m - n)

Q.E.D.