So the statement that I have to prove is as follows $P(n)=n^3 -n$ is divisible by $3$.

Now I have to prove this with backwards induction for all negative numbers but I've already done the same thing but with "straight" induction for all positive numbers.

Do I need to start from the case $P(n+1)$ and go from there to $P(1)$ or $P(-1)$?

Likewise: 

$\begin{align}
P(n+1) & = (n+1)^3 - (n+1) \\
& = n^3 + 3\cdot n^2 + 3 \cdot n + 1 - n + 1 \\
& = 3n(n+1) + n^3 - n 
\end{align}$

$\text{I.H.: So let's suppose the above statement holds for any}$ $n \in \mathbb { Z\_0^{-}} $ 

Then I don't know anymore how to continue. 

For the 2nd part I need to prove that the case of the negative numbers directly follows out of the positive case.

I don't know how to get this part because I'm not sure of the previous case.

Please help me out.