Let - and - be the roots of x2 - 6x - 2 - 0- with - - If an- n- n for n- 1- then the value of a10-2a82a9

We observe that (α +β )(α ^n 1 β ^n 1)=α ^n β ^n+αβ (α ^n 2 β ^n 2) ⇒ 6a_n 1=a_n 2a_n 2 ⇒ 6=a_n 2a_n 2a_n 1 ⇒ 3=a_n 2a_n 22a_n 1, ∀ n gt;2 Putting n =10,we g ...

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Standard XII

Mathematics

Geometric Mean

Let α and β b...

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 $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$

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Solution

The correct option is B 4
We observe that

$(α + β) (α^{n - 1} - β^{n - 1}) = α^{n} - β^{n} + α β (α^{n - 2} - β^{n - 2})$
$\Rightarrow 6 a_{n - 1} = a_{n} - 2 a_{n - 2}$
$\Rightarrow 6 = \frac{a_{n} - 2 a_{n - 2}}{a_{n - 1}}$
$\Rightarrow 3 = \frac{a_{n} - 2 a_{n - 2}}{2 a_{n - 1}}, \forall n > 2$
Putting $n = 10,$ we get

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

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Similar questions

Let α and β be the roots of x²-6x-2 =0 with α>β. If a_n = α_n - β_n for n≥1, then the value of

Q. 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}}

Q. Let

α

and

β

be the roots of equation

x^{2} - 6 x - 2 = 0

. If

a_{n} = α^{n} - β^{n}

, for

n ⩾ 1

, then the value of

\frac{a_{10} - 2 a_{8}}{2 a_{9}}

is equal to

Q. 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}}

Q. Let

α

and

β

be the roots of

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}}{3 a_{9}}

is:

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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

The correct option is B 4We observe that (α+β)(αn−1−βn−1)=αn−βn+αβ(αn−2−βn−2) ⇒6an−1=an−2an−2 ⇒6=an−2an−2an−1 ⇒3=an−2an−22an−1,∀n>2 Putting n=10,we get a10−2a82a9=3

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 $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$