Let the given statement be P(n), i.e., P(n) : n(n + 1)(n + 5) is a multiple of 3. First we prove that the statement is true for n = 1, P(1) : 1(1 + 1) (1 + 5) = 2.6 = 12 and 12 is a multiple of 3. ⇒ P(1) is true. Assume P(k) is true for some positive integer k, i.e., k(k + 1)(k + 5) is a multiple of 3 ... (i) Now we shall prove that P(k + 1) is true i.e., (k + 1)(k + 1 + 1)(k + 1 + 5) is a multiple of 3. For this we have to prove that (k + 1)(k + 2)(k + 6) is a multiple of 3. Let us consider, L.H.S. = (k + 1)(k + 2)(k + 6) = (k + 1)($k^2$ + 8k + 12) = (k + 1) [k2 + 5k + 3k + 12] = (k + 1)[k(k + 5) + 3(k + 4)] = k(k + 1)(k + 5) + 3(k + 1)(k + 4) = 3l + 3(k + 1)(k + 4) (from (i)), where 3l = k(k + 1)(k + 5). = 3[l + (k + 1)(k + 4)] = a multiple of 3. Thus P(k + 1) is true, whenever P(k) is true. Hence, by the principle of mathematical induction P(n) is true ∀ n ∈N.