Question

For all n ≥ 1 , prove that p(n): n(n + 1 )(n + 5) is divisible by 3.

Answer 1

p(1): 1(1 + 1)(1 + 5) = 12divisible by 3, hence true. 

Assuming that true for p(k)

p(k): k(k + 1)(k + 5) is divisible by 3.

k(k + 1)(k + 5) = 3M

p(k + 1): (k + 1)(k + 2)(k + 6)

= (k + 1)(k² + 8k + 12)

= (k + 1)(k² + 5k + 3k +12)

= (k + 1)[k(k + 5) + 3(k + 6)]

= [k(k + 1)(k + 5) + 3(k + 1)(k + 6)]

= [3M + 3(k + 1)(k + 6)]

= 3[M + (k + 1)(k + 6)]

Hence divisible by 3. 

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