Question

Prove that 3 divides $n^3+2n$ whenever n is a positive integer.

Short answer

3 divides $n^3+2n$ whenever n is a positive integer

Step-by-step solution

Step: 1

If n=1,

$1^2+2=3$

it is true for n=1.

Step: 2

Let P(k) be true.

$k^3+2k$

We need to prove that P(k+1) is true.

Step: 3

$\begin{array}{r}(k+1{)}^3+2(k+1)\\ =\left(k^3+2k\right)+\left(3k^2+3k+3\right)\\ =\left(k^3+2k\right)+3\left(k^2+k+1\right)\end{array}$

It is divisible by 3.

It is true for P(k+1) is true.