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

Explanation

Let P(n) =n(n+1) (n+5)
For n=1
P(1)=1.(1+1)(1+5)=12=3(4)
Which is a multiple of 3.
`rArr` p (n) is true for n=1
LetP (n) be true for n=K
`:. P(k) =k(k+1)(K+05)=3lambda "(say) "`
`" Where " lambda in I`
For n=K+1
`P(k+1) =(k+1)(K=2)(K+6)`
`=(K+1)[K^(2)+8K+12]`
`=(k=1)[K^(2)=5k)+(3k+12)]`
`=(k+1)k(k+5)+3(k+1)(+4)`
`=3lamda+3(K+1)(K+4)`
[From equation (1)]
`=3[lambda+(k=1)(K+4)]`
Which is a multiple of 3.
`rArr` P(n) is also true for n=K+1
Hence by the principle of mathematical induction P (n) is true for all natural numbers n.