Question

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

Please give the answers of this question. I hAve seen the solutions but I am not able to understand the last k+1 part.

Answer 1

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

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

let p(k) be 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