Show that `n^3+(n+1)^3+(n+2)^3` is divisible by `9` for every natural number `n`.

Explanation

Let `P(n)=n^(3)+(n+1)^(3)+(n+2)^(3)` is divisible by 9.
When `=1,1^(3),(1+1)^(3)+(1+2)^(3)`
=1+8+27
=36, which is divisible by 9.
`therefore P(1)` is true.
`Rightarrow k^(3)+(k+1)^(3)+(k+2)^(3)` is divisible by 9. (i)
To prove P(k+1) is true i.e.
`(k+1)^(3)+(k+2)^(3)+(k+3)` is divisible by 9.
`=(k+1)^(3)+(k+2)^(3)+k^(3)+9k^(2)+27k+27`
`[k^(3)+(k+1)^(3)+(k+2)^(3)]+9[k^(2)+3k+3]`
`=9m+9(k^(2)+3k+3)`
which is divisible by 9.
Hence P(k+1) is true, whenever P(k) is true.
From (a) and (b), it follows that P(n) is true for all natural number.