Let α, β be roots of x^2 + √2x - 8 = 0. If Un = α^n + β^n, then U10 + √12U9/2U8

Question

Let α, β be roots of \( x^{2}+\sqrt{2} x-8=0\). If \(\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\) then \(\frac{\mathrm{U}_{10}+\sqrt{12} \mathrm{U}_{9}}{2 \mathrm{U}_{8}}\) is equal to _______.

Answer 1

Correct answer is: 4

\(\frac{\alpha^{10}+\beta^{10}+\sqrt{2}\left(\alpha^{9}+\beta^{9}\right)}{2\left(\alpha^{8}+\beta^{8}\right)}\)

\(\frac{\alpha^{8}\left(\alpha^{2}+\sqrt{2} \alpha\right)+\beta^{8}\left(\beta^{2}+\sqrt{2} \beta\right)}{2\left(\alpha^{8}+\beta^{8}\right)}\)

\(\frac{8 \alpha^{8}+8 \beta^{8}}{2\left(\alpha^{8}+\beta^{8}\right)}=4\)