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)(k2 + 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.