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

Prove the following by using the principle of mathematical induction f

1.	Prove the following by using the principle of mathematical induction for all `n in N`:`n(n + 1) (n + 5)`is a multiple of 3.
Answer» Let P(n): n(n+1)(n+5) is a multiple of 3. For n = 1, the given expression becomes `(1xx2xx6) = 12`, which is a multiple of 3. So, the given statement is true for n = 1, i.e., P(1) is true. Let P(k) be true. Then, `P(k) : k(k+5)(k+5)` is a multiple of 3 ` rArr k(k+1)(k+5) = 3m` for some natural number m. ...(i) Now, `(k+1)(k+2)(k+6)` ` =(k+1)(k+2)k+6(k+1)(k+2)` `=k(k+1)(k+5-3)+6(k+1)(k+2)` ` = k(k+1)(k+5)-3k(k+1)+6(k+1)(k+2)` ` = 3m+3(k+1)(2k+4-k)` ` = 3m+3(k+1)(k+4)` ` =3{m+(k+1)(k+4)}`, which is a multiple of 3. ` :. P(k+1):(k+1)(k+2)(k+6)` is a multiple of 3. Thus, P(k+1) is true, whenever P(k) is true. `:. ` P(1) is true and P(k+1) is ture, whenever P(k) is true. Hence , by the principle of mathematical induction, it follows that n(n+1)(n+5) is a multiple of 3 for all ` n in N`.