The integer n3 + 2n is divisible by 3 for every positive integer n

prove it by math induction

is it my proof right?

P (n) = n^3 + 2n is divisible by 3 for every positive integer n.

Let's show that P (n) holds for n = 1

P (1) = 1^3 + 2 (1) = 1 + 2 = 3 which is divisible by 3.

Now assuming, that p (k) is true, let's show that p (k + 1) is also true

p (k + 1) = (k + 1) ^3 + 2 (k + 1) = k^3 + 3k^2 + 3k + 1 + 2k + 2 = k^3 + 3k^2 + 5k + 3 = k^3 + 2k + 3k^2 + 3k + 3 = k^3 + 2k + 3 (k^2 + k + 1), since P (k) is true = > k^3 + 2k is true and 3 (k^2 + k + 1) is divisible by 3 for all values of k wich shows that P (k + 1) is also true.

Therefore, P (n) is true for all positive integer value of n.