Linear Prediction of AR Process
From the definition of the process you know that

$$x_{n+1}=x_n-0.2x_{n-1}+w_{n+1}\tag{1}$$

Since $w_n$ is white you can't predict it, so the best linear predictor for the given process is the filter

$$P(z)=1-0.2z^{-1}\tag{2}$$

which is a first order filter. It estimates the future sample $x_{n+1}$ by computing

$$\hat{x}_{n+1}=x_n-0.2x_{n-1}\tag{3}$$

The residual error is equal to $w_n$, which has an average power of $1$.