For all  n∈N,n(n+1)(n+5) is a multiple of

Let the given statement be P(n).  i.e.  P(n):n(n+1)(n+5) is a multiple of 3.Step l: For n=1,1(1+1)(1+5)=1×2×6=12=3×4which is a multiple of 3, that is trueStep lI: Let it is true for n = k, k(k+1)(k+5)=3λ⇒kk2+5k+k+5=3λ⇒k3+6k2+5k=3λ----iStep III: For n=k+1, (k+1)(k+1+1)(k+1+5)=(k+1)(k+2)(k+6)=k2+2k+k+2(k+6)=k2+3k+2(k+6)=k3+6k2+3k2+18k+2k+12=k3+9k2+20k+12=3λ−6k2−5k+9k2+20k+12  [using Eq. (i) =3λ+3k2+15k+12=3λ+k2+5k+4which is a multiple of 3. Therefore, P(k + 1) is true when P(k) is true. Hence, from the principle of mathematical induction, the statement is true for all natural numbers n.