Another question about large powers of matrices

So the matrix given is...

$\displaystyle A=\left(\begin{array}{cc}1-p & p \\ p & 1-p\end{array}\right)$

...and I've established the eigenvectors are $\displaystyle \left(\begin{array}{c}1 \\ 1\end{array}\right)$ and $\displaystyle \left(\begin{array}{c}1 \\ -1\end{array}\right)$ with corresponding eigenvalues $\displaystyle 1$ and $\displaystyle 1-2p$. I'm then asked to show...

$\displaystyle A^n\left(\begin{array}{c}1 \\ 0\end{array}\right)=\frac{1}{2}\left(\begin{array} {c}1 \\ 1\end{array}\right)+\frac{1}{2}(1-2p)^n\left(\begin{array}{c}1 \\ -1\end{array}\right)$

...and I just can't seem to get it, the method I was taught seems to suggest I need to replace the $\displaystyle \left(\begin{array}{c}1 \\ 0\end{array}\right)$ with the eignevalues and eigenvectors multiplied togeter and then added together, but I still can't quite seem to get that expression.