Let $\alpha$ and $\beta$ be the roots of x² –6x–2 = 0 with $\alpha >\beta$ if ${\mathrm{a}}_{\mathrm{n}}={\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}$ for n>1 then the value of$\frac{a_{10}-2a_8}{2a_9}=$

$$\begin{gathered} \alpha \text{ and }\beta \text{ are the roots of the equation } \\ {x}^2-6x-2=0 \\ {\alpha }^2=6\alpha +2 \\ {\alpha }^{10}=6{\alpha }^9+2{\alpha }^8 \\ {\beta }^{10}=6{\beta }^9+2{\beta }^8 \\ \text{ Subtracting equation (ii) from (i) } \\ \Rightarrow {\alpha }^{10}-{\beta }^{10}=6\left({\alpha }^9-{\beta }^9\right)+2\left({\alpha }^8-{\beta }^9\right) \\ \Rightarrow {\mathrm{a}}_{10}=6{\mathrm{a}}_9+2{\mathrm{a}}_8 \left(\therefore {\mathrm{a}}_{\mathrm{n}}={\mathrm{a}}^{\mathrm{n}}-{\beta }^{\mathrm{n}}\right) \\ \Rightarrow {\mathrm{a}}_{10}-2{\mathrm{a}}_8=6{\mathrm{a}}_9 \\ \Rightarrow \frac{{\mathrm{a}}_{10}-2{\mathrm{a}}_8}{2{\mathrm{a}}_9}=3 \end{gathered}$$