Precalc problem (mathmetical induction)

I haven't done this for years, but I can give it a shot.

Prove true for n = 1, ie prove 1^3 + 2^3 + 3^3 is divisible by 9.
Proof
= 1^3 + 2^3+ 3^3
= 1 + 8 + 27
= 36
= 4*9
Therefore it's true for n = 1

Assume true for n = k, ie assume k^3 + (k+1)^3 + (k+2)^3 = 9M where M in an integer.

Prove true for n = k+1, ie prove (k+1)^3 + (k+2)^3 + (k+3)^3 is divisible by 9.
Proof:
= (k+1)^3 + (k+2)^3 + (k+3)^3
= (k+1)^3 + (k+2)^3 + k^3 + 9k^2 + 27k + 27
= 9M + 9k^2 + 27k + 27
= 9(M+ k^2 + 3k+ 9)

Therefore it's true for n = k+1

Since it's true for n = 1 and it's true for n = k+1, blah blah blah, true for all n.