If  $\alpha ,\beta$  are the roots of the equation  $x^2-ax+b=0$  and  $A_n={\alpha }^n+{\beta }^n$  then which of the following is true? 

 $x^2-ax+b=0$
Both are 
 $$\begin{gathered} {\alpha }^2=a\alpha -b\_\_\left(1\right) \\ {\beta }^2=a\beta -b\_\_\left(2\right) \\ \left(1\right)+\left(2\right) \\ {\alpha }^2+{\beta }^2=a(\alpha +\beta )-b-b \\ multiply\left(1\right){\alpha }^{n-1}and\left(2\right)by\text{\hspace{0.17em}}{\beta }^{n-1} \\ {\alpha }^{n+1}=a{\alpha }^n-b{\alpha }^{n-1} \\ {\beta }^{n+1}=a{\beta }^n-b{\beta }^{n-1} \\ \left(4\right)+\left(5\right) \\ {\alpha }^{n+1}+{\beta }^{n+1} \\ =a({\alpha }^n-{\beta }^n)-b({\alpha }^{n-1}+{\beta }^{n-1}) \\ {A}_{n+1}=a{A}_n-b{A}_{n-1} \end{gathered}$$