Let $\alpha, \beta$ be roots of $x^2+\sqrt{2} x-8=0$. If $\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^n$, then $\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}$ is equal to______

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Aicharya · Accepted Answer

$\begin{aligned} & \frac{\alpha^{10}+\beta^{10}+\sqrt{2}\left(\alpha^9+\beta^9ight)}{2\left(\alpha^8+\beta^8ight)} \ & \frac{\alpha^8\left(\alpha^2+\sqrt{2} \alphaight)+\beta^8\left(\beta^2+\sqrt{2} \betaight)}{2\left(\alpha^8+\beta^8ight)} \ & \frac{8 \alpha^8+8 \beta^8}{2\left(\alpha^8+\beta^8ight)}=4\end{aligned}$

Let $\alpha, \beta$ be roots of $x^2+\sqrt{2} x-8=0$. If…

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