Prove the following using the principle of mathematical induction for all n ∈ N
n (n + 1) (n + 5) is a multiple of 3.

Question

Prove the following using the principle of mathematical induction for all n ∈ N n (n + 1) (n + 5) is a multiple of 3.

Philoid · Accepted Answer

Let the given statement be P(n), as

P(n):n(n + 1)(n + 5) is a multiple of 3.

First, we check if it is true for n = 1,

P(1):1(2)(6) = 12 is a multiple of 3;

∴ It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

P(k):k(k + 1)(k + 5) = 3m where m ∈ N

We shall prove that P(k + 1)is true,

P(k + 1):(k + 1)(k + 2)(k + 5 + 1)

⇒ (k + 1)(k + 2)(k + 5) + (k + 1)(k + 2)

⇒ k(k + 1)(k + 5) + (2)(k + 1)(k + 5) + (k + 1)(k + 2)

⇒ 3m + (k + 1)[2k + 10 + k + 2]

⇒ 3m + (k + 2)(3k + 12)

⇒ 3m + 3(k + 2)(k + 4)

⇒ 3[m + (k + 2)(k + 4)]

We proved that P(k + 1) is true.

Hence by principle of mathematical induction it is true for all n ∈ N.

Prove the following using the principle of mathematical induction for all n ∈ Nn (n + 1) (n + 5) is a multiple of 3.