Mathematical Induction proving

$\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

Re: Mathematical Induction proving

Hint: add $\displaystyle (n+1)^3+2(n+1)-(n^3+2n)=3(n^2+n+1)$ to your induction hypothesis.

Re: Mathematical Induction proving

Same thing:

$\displaystyle (n+1)^3+ 2(n+1)= n^3+ 3n^2+ 3n+ 1+ 2n+ 2= (n^3+ 2n)+ (3n^2+ 3n+ 3)= (n^3+2n)+ 3(n^2+ n+ 1)$.