Question

let alpha and beta are the roots of equation x²-6x-2 = 0.if a_n=alphaⁿ-betaⁿ,for n>,then the value of a₁₀-2a₈/2a₉ =

Answer 1

Dear student
$$\begin{gathered} \mathrm{Since}\mathrm{\alpha }\mathrm{and}\mathrm{\beta }\mathrm{are}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}:{\mathrm{x}}^2-6\mathrm{x}-2=0(\mathrm{\alpha }>\mathrm{\beta })...\left(1\right) \\ \mathrm{If}{\mathrm{a}}_{\mathrm{n}}={\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}\left(\mathrm{n}\ge 1\right) \\ \mathrm{\alpha }\mathrm{is}\mathrm{the}\mathrm{root}\mathrm{of}\mathrm{eq}\left(1\right),\mathrm{therefore} \\ {\mathrm{\alpha }}^2-6\mathrm{\alpha }-2=0 \\ {\mathrm{\alpha }}^2=6\mathrm{\alpha }+2...\left(2\right) \\ \mathrm{Multiplying}\mathrm{through}\mathrm{out}\mathrm{by}{\mathrm{\alpha }}^{\mathrm{n}-2}\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ {\mathrm{\alpha }}^{\mathrm{n}}=6{\mathrm{\alpha }}^{\mathrm{n}-1}+2{\mathrm{\alpha }}^{\mathrm{n}-2}....\left(3\right) \\ \mathrm{Similarly}{\mathrm{\beta }}^{\mathrm{n}}=6{\mathrm{\beta }}^{\mathrm{n}-1}+2{\mathrm{\beta }}^{\mathrm{n}-2}....\left(4\right) \\ \mathrm{Subtracting}\mathrm{eq}\left(4\right)\mathrm{by}\mathrm{eq}\left(3\right),\mathrm{we}\mathrm{get} \\ {\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}=6{\mathrm{\alpha }}^{\mathrm{n}-1}+2{\mathrm{\alpha }}^{\mathrm{n}-2}-6{\mathrm{\beta }}^{\mathrm{n}-1}-2{\mathrm{\beta }}^{\mathrm{n}-2} \\ {\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}=6\left({\mathrm{\alpha }}^{\mathrm{n}-1}-{\mathrm{\beta }}^{\mathrm{n}-1}\right)+2\left({\mathrm{\alpha }}^{\mathrm{n}-2}-{\mathrm{\beta }}^{\mathrm{n}-2}\right) \\ {\mathrm{a}}_{\mathrm{n}}=6{\mathrm{a}}_{\mathrm{n}-1}+2{\mathrm{a}}_{\mathrm{n}-2}....\left(5\right) \\ \mathrm{Now}\mathrm{put}\mathrm{n}=10,\mathrm{we}\mathrm{get} \\ {\mathrm{a}}_{10}=6{\mathrm{a}}_9+2{\mathrm{a}}_8 \\ \mathrm{Therefore},{\mathrm{a}}_{10}-2{\mathrm{a}}_8=6{\mathrm{a}}_9 \\ \mathrm{hence} \\ \frac{{\mathrm{a}}_{10}-2{\mathrm{a}}_8}{2{\mathrm{a}}_9}=3 \end{gathered}$$
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