Question

For all n ≥ 1, prove that p(n): n³ + (n + 1)³ + (n + 2)³ is divisible by 9.

Answer 1

p(1): 1 + 2³ + 3³ = 1 + 8 + 27 = 36 divisible by 9, hence true. 

Assuming that true for p(k)

p(k): k³ + (k + 1)³ + (k + 2)³ is divisible by 9.

⇒ k³ + (k + 1)³ + (k + 2)³ = 9M

p(k +1 ): (k + 1)³ + (k + 2 )³ + (k + 3)³

= (k +1)³ + (k + 2)³ + k³ + 9k² + 27k + 27

= [(k + 1)³ + (k + 2)³ + k³] + 9[k² + 3k + 3]

= 9M + 9[k² + 3k + 3]

Hence p(k + 1)divisible by 9. 

Therefore by using the principle of mathematical induction true for all n ∈ N.