Question

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

Answer 1

it can be prove by principal of mathematical indication

Answer 2

p(k): k (k+1)(k+5) = 3x where x € N
= (k^2 +k)(k+5)=3x
= k^3 + 5k^2 + 5k + k^2 =3x
= k^3 + 6k^2 + 5k = 3x
{ k^3= 3x - 6k^2 -5k} -----(i)

To prove,
p(k+1): (k+1)(k+2)(k+6) =3x where x € N
i.e,
p(k+1) : (k+1)(k+2)(k+6)
= k(k+2)(k+6) + (k+2)(k+6)
= k(k^2 +6k +2k +12) + k^2 + 8k + 12
= k^3 + 8k^2 +12k + k^2 + 8k + 12
= k^3 + 9k^2 + 20k +12
= 3x - 6k^2 -5k + 9k^2 + 20k + 12
= 3x + 3k^2 + 15k + 12
= 3( x + k^2 + 5k + 4)
Hence proved multiple of 3.