Let - and - be the roots of equation x2-6x-2-0- If an -n-n- for n - 1- then the value of a10-2a8-2a9 is

Given, α and β are the roots of the equation x^2 6x 2=0. ∴ a_n =a^n β^n for n ≥ 1 a_10=α^10 β^10 a_8 =α^8 β^8 a_9 =α^9 β^9 Now, consider a_10 2a_8/2a_9=α^ ...

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Byju's Answer

Standard XIII

Mathematics

Relation between Roots and Coefficients for Quadratic

Let α and β b...

Question

Let $α$ and $β$ be the roots of equation $x^{2} - 6 x - 2 = 0$ . 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

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Solution

The correct option is C 3
Given, $α$ and $β$ are the roots of the equation $x^{2} - 6 x - 2 = 0$ .
$∴ a_{n} = a^{n} - β^{n} f o r n \geq 1 a_{10} = α^{10} - β^{10} a_{8} = α^{8} - β^{8} a_{9} = α^{9} - β^{9}$
Now, consider
$\frac{a_{10} - 2 a_{8}}{2 a_{9}} = \frac{α^{10} - β^{10} - 2 (α^{8} - β^{8})}{2 (α^{9} - α^{9})} = \frac{α^{8} (α^{2} - 2) - β^{8} (β^{2} - 2)}{2 (α^{9} - β^{9})} = \frac{α^{8} .6 α - β^{8} 6 β}{2 (α^{9} - β^{9})} = \frac{6 α^{9} - 6 β^{9}}{2 (α^{9} - 6 β^{9})} = \frac{6}{2} = 3$
$⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} ∵ α a n d β a r e t h e r o o t s o f x^{2} - 6 x - 2 = 0 o r x^{2} = 6 x + 2 \Rightarrow α^{2} = 6 α + 2 \Rightarrow α^{2} - 2 = 6 α a n d β^{2} = 6 β + 2 \Rightarrow β^{2} - 2 = 6 β \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
Alternate Solution
Since, $α$ and $β$ are the roots of the equation
$x^{2} - 6 x - 2 = 0$
or $x^{2} = 6 x + 2$
$∴ α^{2} = 6 α + 2 \Rightarrow α^{10} = 6 α^{9} + 2 α^{8} . . . . (i)$
Similarly, $β^{10} = 6 β^{9} + 2 β^{8} . . . . (i i)$
On subtracting Eq. (ii) from Eq. (i), we get
$α^{10} - β^{10} = 6 (α^{9} - β^{9}) + 2 (α^{8} - β^{8}) (∵ a_{n} = α^{n} - β^{n}) \Rightarrow a_{10} = 6 a_{9} + 2 a_{8} \Rightarrow a_{10} - 2 a_{8} = 6 a_{9} \Rightarrow \frac{a_{10} - 2 a_{8}}{2 a_{9}} = 3$

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Let α and β be the roots of equation x2−6x−2=0. If an=αn−βn, for n≥1, then the value of a10−2a82a9 is

Let $α$ and $β$ be the roots of equation $x^{2} - 6 x - 2 = 0$ . 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