Let and be the roots of x 2 - 6 x - 2 = 0 , with . If a n = n - n for n 1 , then the value of a 10 - 2 a 8 2 a 9 is [2011]

Question

Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0$ , with $α > β$ . If $a_{n} = α^{n} - β^{n} for n \geq 1,$ then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is [2011]

Smart Achievers · Accepted Answer

(3)

$∵$ $α, β are the roots of x^{2} - 6 x - 2 = 0$

$a_{n} = α^{n} - β^{n} for n \geq 1,$ $α^{2} - 6 α - 2 = 0$

$\Rightarrow α^{10} - 6 α^{9} - 2 α^{8} = 0$

$\Rightarrow α^{10} - 2 α^{8} = 6 α^{9} \dots (i)$

$Similarly β^{10} - 2 β^{8} = 6 β^{9} \dots (i i)$

$On subtracting (ii) from (i),$

$α^{10} - β^{10} - 2 (α^{8} - β^{8}) = 6 (α^{9} - β^{9})$

$\Rightarrow a_{10} - 2 a_{8} = 6 a_{9}$

$\Rightarrow \frac{a_{10} - 2 a_{8}}{2 a_{9}} = 3$

Solution

Q.
Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0$ , with $α > β$ . If $a_{n} = α^{n} - β^{n} for n \geq 1,$ then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is [2011]

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Solution

Q. Let α and β be the roots of x2-6x-2=0, with α>β. If an=αn-βn for n≥1, then the value of a10-2a82a9 is [2011]

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Ans. (3) ∵ α,β are the roots of x2-6x-2=0 ∴ α2-6α-2=0 ⇒α10-6α9-2α8=0 ⇒α10-2α8=6α9 ⋯(i) Similarly β10-2β8=6β9 ⋯(ii) On subtracting (ii) from (i), α10-β10-2(α8-β8)=6(α9-β9) ⇒a10-2a8=6a9 ⇒a10-2a82a9=3

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Q.
Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0$ , with $α > β$ . If $a_{n} = α^{n} - β^{n} for n \geq 1,$ then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is [2011]