$\begin{aligned}
& \mathrm{x}^2-\sqrt{2 \mathrm{x}}-\sqrt{3}=0\left\langle_\beta^\alpha\right. \
& \alpha^{\mathrm{n}+2}-\sqrt{2} \alpha^{\mathrm{n}+1}-\sqrt{3} \alpha^{\mathrm{n}}=0 \
& \text { and } \beta^{\mathrm{n}+2}-\sqrt{2} \beta^{\mathrm{n}+1}-\sqrt{3} \beta^{\mathrm{n}}=0
\end{aligned}$
Subtracting $\begin{aligned}
& \left(\alpha^{\mathrm{n}+2}-\beta^{\mathrm{n}+2}\right)-\sqrt{2}\left(\alpha^{\mathrm{n}+1}-\beta^{\mathrm{n}+1}\right)-\sqrt{3}\left(\alpha^{\mathrm{n}}-\beta^{\mathrm{n}}\right)=0 \
& \Rightarrow \mathrm{P}{\mathrm{n}+2}-\sqrt{2} \mathrm{P}{\mathrm{n}+1}-\sqrt{3} \mathrm{P}{\mathrm{n}}=0
\end{aligned}\begin{aligned} & \text { Put } \mathrm{n}=10 \ & \mathrm{P}{12}-\sqrt{2} \mathrm{P}{11}-\sqrt{3} \mathrm{P}{10}=0 \ & \mathrm{n}=9 \ & \mathrm{P}{11}-\sqrt{2} \mathrm{P}{10}-\sqrt{3} \mathrm{P}9=0 \ & 11\left(\sqrt{3} \cdot \mathrm{P}{10}+\sqrt{2} \mathrm{P}{11}-\mathrm{P}{11}\right)-10\left(\sqrt{2} \mathrm{P}{10}-\mathrm{P}{11}\right) \ & =0-10\left(-\sqrt{3} \mathrm{P}_9\right)=10 \sqrt{3} \mathrm{P}_9\end{aligned}$