Let p - q be integers and let - - be the roots of the equation x 2-2 x -3-0 where - - For n -0-1-2- - let a n - p - n - q - n- thena9-A- 3 a8-5 a7-3 a6B- 2 a 8-3 a 7- 4 a 8-2 a 7-3 a 6D- 2 a 8-3 a 6

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Roots of a Quadratic Equation

Let p , q be ...

Question

Let $p, q$ be integers and let $α, β$ be the roots of the equation $x^{2} - 2 x + 3 = 0$ where $α \neq β .$ For $n = 0, 1, 2, . . .,$ let $a_{n} = p α^{n} + q β^{n},$ then $a_{9} =$

$2 a_{8} + 3 a_{7}$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

$3 a_{8} - 5 a_{7} + 3 a_{6}$

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

$4 a_{8} + 2 a_{7} - 3 a_{6}$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

$2 a_{8} + 3 a_{6}$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B
$3 a_{8} - 5 a_{7} + 3 a_{6}$

Since, $α$ and $β$ are the roots of the equation, $x^{2} - 2 x + 3 = 0$ ,

so, $α + β = 2$ and $α β = 3$

Also, $α^{2} - 2 α + 3 = 0$ and $β^{2} - 2 β + 3 = 0$

$\Rightarrow α^{2} = 2 α - 3$ and $β^{2} = 2 β - 3$

Given, $a_{n} = p α^{n} + q β^{n}$

So, $a_{0} = p + q$

$a_{1} = p α + q β$

$a_{2} = p α^{2} + q β^{2}$

$= p (2 α - 3) + q (2 β - 3)$

$= 2 (p α + q β) - 3 (p + q)$

$= 2 a_{1} - 3 a_{0}$

Similarly, $a_{n + 2} = 2 a_{n + 1} - 3 a_{n}$ ... (1)

and $a_{n + 3} = 2 a_{n + 2} - 3 a_{n + 1}$ ... (2)

Subtracting equation (1) from (2), we get

$a_{n + 3} = 3 a_{n + 2} - 5 a_{n + 1} + 3 a_{n}$

Thus, $a_{9} = 3 a_{8} - 5 a_{7} + 3 a_{6}$

Suggest Corrections

Similar questions

Q. Let

p, q

be integers and

α, β

be the roots of the equation

x^{2} - 2 x + 3 = 0

where

α \neq β

If a_{n} = p α^{n} + q β^{n} where n \in {0, 1, 2, . . . . .}

, then the value of

a_{9}

Let $p, q$ be integers and let $α, β$ be the roots of the equation $x^{2} - 2 x + 3 = 0$ where $α \neq β .$ For $n = 0, 1, 2, . . .,$ let $a_{n} = p α^{n} + q β^{n}$ and $a_{3} = - 10,$ then $p^{q} + q^{p} =$ ___

Q. Let

p, q

be integers and let

α, β

be the roots of the equation,

x^{2} - x - 1 = 0

, where

α \neq β

. For

n = 0, 1, 2, . . .

let

a_{n} = p α^{n} + q β^{n}

.
FACT : If

a

and

b

are rational numbers and

a + b \sqrt{5} = 0

, then

a = 0 = b

a_{12} =

Q. Let

p, q

be integers and

α, β

be the roots of the equation

x^{2} - 2 x + 3 = 0

where

α \neq β

If a_{n} = p α^{n} + q β^{n} where n \in {0, 1, 2, . . . . .}

, then the value of

a_{9}

Q. Let

p, q

be integers and let

α, β

be the roots of the equation,

x^{2} - x - 1 = 0,

where

α \neq β .

For

n = 0, 1, 2, . . . .,

let

a_{n} = p α^{n} + q β^{n} .

FACT: If

a

and

b

are rational numbers and

a + b \sqrt{5} = 0.

then

a = 0 = b .

a_{4} = 28,

then

p + 2 q =

Join BYJU'S Learning Program

Let p, q be integers and let α,β be the roots of the equation x2–2x+3=0 where α≠β. For n=0, 1, 2,..., let an=pαn+qβn, then a9=

Let $p, q$ be integers and let $α, β$ be the roots of the equation $x^{2} - 2 x + 3 = 0$ where $α \neq β .$ For $n = 0, 1, 2, . . .,$ let $a_{n} = p α^{n} + q β^{n},$ then $a_{9} =$