If α , β are roots of the equation x2+5(2)x+10=0 , the value of (P17P20+52P17P19P18P19+52P182) α>β,Pn=αn−βn,n∈N is equal to _________

α,β are roots of the equationx2+5(2)x+10=0,α>β Pn=αn−βn , where n is a positive integer,Now, P17[P20+52P19]P18[P19+52P18] =P17P18[(α20−β20)+52(α19−β19)(α19−β19)+52(α18−β18)] =P17P18[10β18−10α1810β17−10α17]=P17P18(−10⋅P18)(−10⋅P17)=1

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

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Detailed Solution

$α, β$ are roots of the equation
$x^{2} + 5 (\sqrt{2}) x + 10 = 0, α > β$
$P_{n} = α^{n} - β^{n}$ , where n is a positive integer,
Now,
$\begin{array}{r} \frac{P_{17} [P_{20} + 5 \sqrt{2} P_{19}]}{P_{18} [P_{19} + 5 \sqrt{2} P_{18}]} \\ = \frac{P_{17}}{P_{18}} [\frac{(α^{20} - β^{20}) + 5 \sqrt{2} (α^{19} - β^{19})}{(α^{19} - β^{19}) + 5 \sqrt{2} (α^{18} - β^{18})}] \\ = \frac{P_{17}}{P_{18}} [\frac{10 β^{18} - 10 α^{18}}{10 β^{17} - 10 α^{17}}] = \frac{P_{17}}{P_{18}} \frac{(- 10 \cdot P_{18})}{(- 10 \cdot P_{17})} = 1 \end{array}$