SOLUTION: prove n^3 + 2n is divisible by 3 induction

    At n= 1  n^3 + 2n = 1^3 + 2*1 = 3  is divisible by 3.

    Thus the base of induction is valid.

    Let assume that  P(n) = n^3 + 2n is divisible by 3,



    Then  P(n+1) = (n+1)^3 + 2*(n+1) = n^3 + 3n^2 + 3n + 1 + 2n + 2 = 

                 = (re-group) = (n^3 + 2n) + (3n^2 + 3n + 3) = P(n) + 3*(n^2 + n +1).



    So, P(n+1) is the sum of P(N) and the other addend, which is multiple of 3.


    Thus, if P(n) is divisible by n, then P(n+1) is divisible by 3, too.


    The inductive step is proven to be valid.