If α,β are the roots of the equation x2+52x+10=0,α>β and Pn=αn−βn for each positive integer n then P17.P20+52 P17.P19P18.P19+52 P182=___

α,β are roots of x2+52x+10=0 α2+52α+10=0 multiply with αn−2on both sides αn+53αn−1+10αn−2=0−(1) βn+52βn++10.βn−2=0−(2)by addy (1)&(2)(αn+βn)+52(αn++βn+)+10(αn−2+βn−2) ⇒Pn+52Pn−1+10.Pn−2=0 forn=20⇒P20+52P19+10.P18=0 n=19⇒P19+52P18+10.P17=0∴ =Required=P17[P20+52P19]P18[P19+52P18]=P17.[−10.P18]P18[−10.P17]=1

If α,β are the roots of the equation x2+52x+10=0,α>β and Pn=αn−βn for each positive integer n then P17.P20+52 P17.P19P18.P19+52 P182=___

α,β are roots of x2+52x+10=0 α2+52α+10=0 multiply with αn−2on both sides αn+53αn−1+10αn−2=0−(1) βn+52βn++10.βn−2=0−(2)by addy (1)&(2)(αn+βn)+52(αn++βn+)+10(αn−2+βn−2) ⇒Pn+52Pn−1+10.Pn−2=0 forn=20⇒P20+52P19+10.P18=0 n=19⇒P19+52P18+10.P17=0∴ =Required=P17[P20+52P19]P18[P19+52P18]=P17.[−10.P18]P18[−10.P17]=1

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If $α, β$ are the roots of the equation $x^{2} + 5 \sqrt{2} x + 10 = 0, α > β$ and $P_{n} = α^{n} - β^{n}$ for each positive integer n then $\frac{P_{17.} P_{20} + 5 \sqrt{2} P_{17} . P_{19}}{P_{18} . P_{19} + 5 \sqrt{2} {P_{18}}^{2}} =___$

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Detailed Solution

$\begin{array}{l} α, β a r e r o o t s o f x^{2} + 5 \sqrt{2} x + 10 = 0 \\ α^{2} + 5 \sqrt{2} α + 10 = 0 \\ m u l t i p l y w i t h α^{n - 2} o n b o t h s i d e s \\ α^{n} + 5 \sqrt{3} α^{n - 1} + 10 α^{n - 2} = 0 - (1) \\ β^{n} + 5 \sqrt{2} β^{n +} + 10. β^{n - 2} = 0 - (2) \\ b y a d d y (1) & (2) (α^{n} + β^{n}) + 5 \sqrt{2} (α^{n +} + β^{n +}) + 10 (α^{n - 2} + β^{n - 2}) \\ \Rightarrow P_{n} + 5 \sqrt{2} P_{n - 1} + 10. P_{n - 2} = 0 \\ f o r n = 20 \Rightarrow P_{20} + 5 \sqrt{2} P_{19} + 10. P_{18} = 0 \\ n = 19 \Rightarrow P_{19} + 5 \sqrt{2} P_{18} + 10. P_{17} = 0 \\ ∴ = Re q u i r e d = \frac{P_{17} [P_{20} + 5 \sqrt{2} P_{19}]}{P_{18} [P_{19} + 5 \sqrt{2} P_{18}]} = \frac{P_{17} . [- 10. P_{18}]}{P_{18} [- 10. P_{17}]} = 1 \end{array}$