$\displaystyle n^{3}+2n $

prove that it is divisible by 3 for any integer $\displaystyle n\geq 1 $

the steps i use are

1) plug in a number to see if its true 1+2 = 3 is divisible by 3= true

2). K=N plug it in $\displaystyle k^{3}+2k $

3) K+1 =N plug it in $\displaystyle k^{3}+2k+ (k+1)^{3} +2(k+1) $ ---- this is where I'm confused i usually have numbers for example 1+3+4+5+.......(3n+2) = n(n+1)

I'm not sure if step 3 is right because that's what i do for my usual example with numbers