If \( \alpha, \beta \) are roots of the equation P \( x^{2}+5(\sqrt...

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If \( \alpha, \beta \) are roots of the equation
P \( x^{2}+5(\sqrt{2}) x+10=0, \alpha\beta \) and \( P_{n}=\alpha_{n}-\beta_{n} \) for

W each positive integer \( n \), then the value of \( \left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{17} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right) \) is equal to
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