If - and - are the roots of the equation x2-a x-b-0 and An-n-n- then which of the following is true-A- An-1-a An-b An-1B- An-1-b An-b An-1C- An-1-b An-a An-1D- An-1-a An-b An-1

The correct option is D A_n+1= aA_n bA_n 1 nbsp; nbsp;A_n+1 = α^n+1 + β^n+1 = α^n+1 + α^n β + β^n+1 +αβ^n α^n β αβ^n = α^n (α + β ) + β^n (β +α) αβ ( ...

211

You visited us 211 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XIII

Mathematics

Relation between Roots and Coefficients for Quadratic

If α and β ar...

Question

If $α$ and $β$ are the roots of the equation $x^{2} - a x + b = 0$ and $A_{n} = α^{n} + β^{n}$ , then which of the following is true?

A_{n + 1} = a A_{n} + b A_{n - 1}

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

A_{n + 1} = b A_{n} + b A_{n - 1}

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

A_{n + 1} = a A_{n} - b A_{n - 1}

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

A_{n + 1} = b A_{n} - a A_{n - 1}

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

Open in App

Solution

The correct option is C $A_{n + 1} = a A_{n} - b A_{n - 1}$
$A_{n + 1} = α^{n + 1} + β^{n + 1}$
$= α^{n + 1} + α^{n} β + β^{n + 1} + α β^{n} - α^{n} β - α β^{n}$
$= α^{n} (α + β) + β^{n} (β + α) - α β (α^{n - 1} + β^{n - 1})$
$= (α + β) (α^{n} + β^{n}) - α β (α^{n - 1} + β^{n - 1})$
$= a A_{n} - b A_{n - 1}$

Suggest Corrections

Similar questions

Q. If

α, β

are the roots of the equation

x^{2} - a x + b = 0

and

A_{n} = α^{n} + β^{n}

, then which of the following is true?

Q. If

α

and

β

are the roots of the equations

x^{2} - a x + b = 0

and

A_{n} = α^{n} + β^{n}

, then which of the following is true?

$I f α, β are the roots of the equation x^{2} - x - 1 = 0 and A_{n} = α^{n} + β^{n} then A_{n + 2} + A_{n - 2} = - -$

Q. Let

α

and

β

be the roots of

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

with

α

β

, For all positive integers n, define,

a_{n} = \frac{α^{n} - β^{n}}{α - β}, n \geq 1

b_{1} = 1

and

b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2

.
Then which of the following option is/are correct?

Q. Let

α

and

β

be the roots of

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

with

α

β

, For all positive integers n, define,

a_{n} = \frac{α^{n} - β^{n}}{α - β}, n \geq 1

b_{1} = 1

and

b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2

.
Then which of the following option is/are correct?

Join BYJU'S Learning Program

If α and β are the roots of the equation x2−ax+b=0 and An=αn+βn, then which of the following is true?

The correct option is C An+1=aAn−bAn−1 An+1=αn+1+βn+1 =αn+1+αnβ+βn+1+αβn−αnβ−αβn =αn(α+β)+βn(β+α)−αβ(αn−1+βn−1) =(α+β)(αn+βn)−αβ(αn−1+βn−1) =aAn−bAn−1

If $α$ and $β$ are the roots of the equation $x^{2} - a x + b = 0$ and $A_{n} = α^{n} + β^{n}$ , then which of the following is true?