If  $\alpha , \beta$  are roots of the equation  $x^2+5\left(\sqrt{2}\right)x+10=0$ , the value of  $\left(\frac{P_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{P_{18}{P}_{19}+5\sqrt{2}{P}_{18}^2}\right)$  $\alpha >\beta ,P_n={\alpha }^n-{\beta }^n,n\in N$ is equal to _________ 

$\alpha ,\beta$  are roots of the equation
$x^2+5\left(\sqrt{2}\right)x+10=0,\alpha >\beta$
  $P_n={\alpha }^n-{\beta }^n$ , where n is a positive integer,
Now,
$\begin{array}{rr} & \frac{P_{17}[P_{20}+5\sqrt{2}{P}_{19}]}{P_{18}[P_{19}+5\sqrt{2}{P}_{18}]}\\ & =\frac{P_{17}}{P_{18}}\left[\frac{({\alpha }^{20}-{\beta }^{20})+5\sqrt{2}({\alpha }^{19}-{\beta }^{19})}{({\alpha }^{19}-{\beta }^{19})+5\sqrt{2}({\alpha }^{18}-{\beta }^{18})}\right]\\ & =\frac{P_{17}}{P_{18}}\left[\frac{10{\beta }^{18}-10{\alpha }^{18}}{10{\beta }^{17}-10{\alpha }^{17}}\right]=\frac{P_{17}}{P_{18}}\frac{(-10\cdot P_{18})}{(-10\cdot P_{17})}=1\\ & \end{array}$