First slide

Theory of expressions

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

Moderate

A

1
B

2
C

3
D

4

Solution

$a_{n} = α^{n} - β^{n}$
Also $α^{2} - 6 α - 2 = 0$
Multiply with $α^{8}$ on both sides
$\Rightarrow α^{10} - 6 α^{9} - 2 α^{8} = 0$ …… (1)
similarly $β^{10} - 6 β^{9} - 2 β^{8} = 0$ ……. (2)
Subtracting (2) from (1) we have
$\begin{array}{l} α^{10} - β^{10} - 6 (α^{9} - β^{9}) = 2 (α^{8} - β^{8}) \\ \Rightarrow a_{10} - 6 a_{9} = 2 a_{8} \\ \Rightarrow \frac{a_{10} - 2 a_{8}}{2 a_{9}} = 3 . \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App