If  $\alpha ,\beta$  are roots of Quadratic equation  $x^2+5\sqrt{2}x+10=0,\alpha >\beta$  and  $P_n={\alpha }^n-{\beta }^n$  for each positive integer n then value of  $\frac{P_{17}.P_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{P_{18}{P}_{19}+5\sqrt{2}{P}_{18}^2}$  is

$\because \alpha ,\beta$   are roots of  $x^2+5\sqrt{2}x+10=0$
$a+\beta =-5\sqrt{2} P_n={\alpha }^n-{\beta }^n$   
Now   $\frac{P_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{P_{18}{P}_{19}+5\sqrt{2}{P}_{18}^2}\Rightarrow \frac{P_{17}(P_{20}+5\sqrt{2}{P}_{19})}{P_{18}(P_{19}+5\sqrt{2}{P}_{18})}$
$$\begin{gathered} \Rightarrow \frac{P_{17}}{P_{18}}.\frac{({\alpha }^{20}-{\beta }^{20}+5\sqrt{2}({\alpha }^{19}-{\beta }^{19}))}{({\alpha }^{19}-{\beta }^{19}+5\sqrt{2}({\alpha }^{18}-{\beta }^{18}))} \\ \begin{array}{l}\Rightarrow \frac{P_{17}}{P_{18}}.\frac{({\alpha }^{19}(\alpha +5\sqrt{2})-{\beta }^{19}(\beta +5\sqrt{2}))}{({\alpha }^{18}(\alpha +5\sqrt{2})-{\beta }^{19}(\beta +5\sqrt{2}))}\\ \Rightarrow \frac{P_{17}}{P_{18}}.\frac{[{\alpha }^{19}(-\beta )-{\beta }^{19}(-\alpha )]}{[{\alpha }^{18}(-\beta )-{\beta }^{18}(-\alpha )]}\\ \Rightarrow \frac{P_{17}}{P_{18}}.\frac{\alpha \beta ({\alpha }^{18}-{\beta }^{18})}{\alpha \beta ({\alpha }^{17}-{\beta }^{17})}\Rightarrow \frac{P_{17}}{P_{18}}.\frac{P_{18}}{P_{17}}=1\end{array} \end{gathered}$$