Add.Maths - MI problems(division)

2n^3+n is divisible by 3 for all positive integers n.
When n=1, 2(1)^3+1=3 which is divisible by 3.
Assume it si true when n=k, i.e.,2k^3+k=3M ,where M is an integer.
Consider n=k+1,
2(k+1)^3+(k+1)
=(k+1)[2(k^2+2k+1)+1]
=2k^3+4k^2+3k+2k^2+4k+3
=3M-k+6k^2+7k+3
=3(M+2k^2+2k+1) , where M+2k^2+2k+1 is an integer.
Therefore, it is true when n=k+1
By the principle of mathematical induction, the statment is true for all positive integers n.