Question

Let $$\alpha$$ and $$\beta$$ be the roots of equation $$x^2-6x-2=0$$. If $$a_n=\alpha^n-\beta^n$$, for $$n\geqslant 1$$, then the value of $$\dfrac {a_{10}-2a_8}{2a_9}$$ is equal to

A
6
B
6
C
3
D
3

Solution

The correct option is C $$3$$$$\because \alpha, \beta$$ are the roots$$\Rightarrow \alpha^2-6\alpha-2=0\Rightarrow \alpha^2-2=6\alpha$$and $$\beta^2-6\beta-2=0\Rightarrow \beta^2-2=6\beta$$Hence, $$\cfrac {a_{10}-2a_8}{2a_9}=\cfrac {\alpha^{10}-\beta^{10}-2(\alpha^{8}-\beta^{8})}{2(\alpha^{9}-\beta^{9})}$$$$=\cfrac {\alpha^8(\alpha^2-2)-\beta^{8}(\beta^2-2)}{2(\alpha^{9}-\beta^{9})}=\cfrac {\alpha^8.6\alpha-\beta^{8}.6\beta}{2(\alpha^{9}-\beta^{9})}=\cfrac {6(\alpha^9-\beta^{9})}{2(\alpha^{9}-\beta^{9})}=3$$ Mathematics

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