Show using the principle of mathematical induction that n(n+1)(n+5) is

1.	Show using the principle of mathematical induction that n(n+1)(n+5) is a multiple of 3 .
Answer» 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 itsamultiple 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^3from (1) =3y-6k^2-5k+9k^2+20k +12 =3y+3k^2+15k +12 3(y+k^25k +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

Show using the principle of mathematical induction that n(n+1)(n+5) is a multiple of 3 .

Answer»

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 itsamultiple 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^3from (1)

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

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

3(y+k^25k +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