Prove that n(n+1)(n+5) is a multiple of 3

p(n)=n(n+1)(n+5)

p(1)=1(1+1)(1+5)=1x2x6=12

let p(k) b true

p(k)=k(k+1)(k+5)

on solving

=k^{3}+6k^{2}+5k=3y( as its^{ }a multiple of 3)

k^{3}=3y-6k^{2}-5k ------(1)

p(k+1)= k+1 ( k+1 +1) k+1 +5)

=k+1 ( k+2) k+6)

on solving =k^{3}+9k^{2}+20k +12

putting value of k^{3} from (1)

=3y-6k^{2}-5k+9k^{2}+20k +12

=3y+3k^{2}+15k +12

3(y+k^{2}5k +4)

therefore we can say p(k+1) is a multiple of 3

p(k) is true so p(n) is also true and its a multiple of 3