# If α, β are roots of the equation

Question:

If $\alpha, \beta$ are roots of the equation $\mathrm{x}^{2}+5(\sqrt{2}) \mathrm{x}+10=0, \alpha>\beta$ and $\mathrm{P}_{\mathrm{n}}=\alpha^{\mathrm{n}}-\beta^{\mathrm{n}}$ for each positive integer $\mathrm{n}$, then the value of

$\left(\frac{\mathrm{P}_{17} \mathrm{P}_{20}+5 \sqrt{2} \mathrm{P}_{17} \mathrm{P}_{19}}{\mathrm{P}_{18} \mathrm{P}_{19}+5 \sqrt{2} \mathrm{P}_{18}^{2}}\right)$ is equal to__________.

Solution:

$x^{2}+5 \sqrt{2} x+10=0$

$\& \mathrm{p}_{\mathrm{n}}=\alpha^{\mathrm{n}}-\beta^{\mathrm{n}}$ (Given)

Now $\frac{\mathrm{P}_{17} \mathrm{P}_{20}+5 \sqrt{2} \mathrm{P}_{17} \mathrm{P}_{19}}{\mathrm{P}_{18} \mathrm{P}_{19}+5 \sqrt{2} \mathrm{P}_{18}^{2}}=\frac{\mathrm{P}_{17}\left(\mathrm{P}_{20}+5 \sqrt{2} \mathrm{P}_{19}\right)}{\mathrm{P}_{18}\left(\mathrm{P}_{19}+5 \sqrt{2} \mathrm{P}_{18}\right)}$

$\frac{P_{17}\left(\alpha^{20}-\beta^{20}+5 \sqrt{2}\left(\alpha^{19}-\beta^{19}\right)\right)}{P_{18}\left(\alpha^{19}-\beta^{19}+5 \sqrt{2}\left(\alpha^{18}-\beta^{18}\right)\right)}$

$\frac{P_{17}\left(\alpha^{19}(\alpha+5 \sqrt{2})-\beta^{19}(\beta+5 \sqrt{2})\right)}{P_{18}\left(\alpha^{18}(\alpha+5 \sqrt{2})-\beta^{18}(\beta+5 \sqrt{2})\right)}$

Since $\alpha+5 \sqrt{2}=-10 / \alpha$......(1)

$\& \beta+5 \sqrt{2}=-10 / \beta$.......(2)

Now put there values in above expression

$=-\frac{10 P_{17} P_{18}}{-10 P_{18} P_{17}}=1$

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