Let and be the roots of the equation x 2 - 6 x - 2 = 0 . If a n = n - n , for n 1 , then the value of a 10 - 2 a 8 2 a 9 is equal to

Question

Let $α$ and $β$ be the roots of the 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 equal to

Smart Achievers · Accepted Answer

(1)

$α is a root of x^{2} - 6 x - 2 = 0$

$Then α^{2} - 6 α - 2 = 0$

Multiplying by $α^{n}$ , it becomes,

$α^{n + 2} - 6 α^{n + 1} - 2 α^{n} = 0$ ...(i)

Similarly, $β^{n + 2} - 6 β^{n + 1} - 2 β^{n} = 0$ ...(ii)

Subtracting (ii) from (i), we get

$(α^{n + 2} - β^{n + 2}) - 6 (α^{n + 1} - β^{n + 1}) - 2 (α^{n} - β^{n}) = 0$

$i.e., a_{n + 2} - 6 a_{n + 1} - 2 a_{n} = 0$

Thus, $\frac{a_{n + 2} - 2 a_{n}}{2 a_{n + 1}} = 3$

Set $n = 8$ to obtain the desired value $\frac{a_{10} - 2 a_{8}}{2 a_{9}} = 3$

Solution

Q.
Let $α$ and $β$ be the roots of the 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 equal to

1 3

2 - 3

3 6

4 - 6

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Solution

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

1 3

2 - 3

3 6

4 - 6

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Q.
Let $α$ and $β$ be the roots of the 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 equal to