Let - and - be the roots of x2-x-1-0- If n be positive integer- then -n-n isA- 2 cos2 n -3C- 2 cos n x -3 -D- 2 sin n -3

X^2+x+1=0 ⇒ x= 1±√( 3)2 Let α = 1+ √(3) i2, β = nbsp; 1 √(3) i2 ⇒α =exp( 2π i3), β = nbsp;exp( 2π i3) α^n +β^n=exp( 2nπ i3), β = nbsp;exp( 2nπ i3) n ...

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

Byju's Answer

Standard XII

Mathematics

Polar Representation of a Complex Number

Let α and β b...

Question

Let $α$ and $β$ be the roots of $x^{2} + x + 1 = 0$ . If $n$ be positive integer, then $α^{n} + β^{n}$ is

2 cos \frac{2 n π}{3}

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

2 sin \frac{2 n π}{3}

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

2 cos \frac{n π}{3}

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

2 sin \frac{n π}{3}

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

Open in App

Solution

The correct option is A $2 cos \frac{2 n π}{3}$
$x^{2} + x + 1 = 0 \Rightarrow x = \frac{- 1 \pm \sqrt{- 3}}{2}$
Let $α = \frac{- 1 + \sqrt{3} i}{2}, β = \frac{- 1 - \sqrt{3} i}{2}$
$\Rightarrow α = exp (\frac{2 π i}{3}), β = exp (\frac{- 2 π i}{3})$

$α^{n} + β^{n} = exp (\frac{2 n π i}{3}), β = exp (\frac{- 2 n π i}{3})$
$= 2 cos \frac{2 n π}{3} [∵ e^{i θ} = cos θ + i sin θ]$

Suggest Corrections

Similar questions

Q. Let

α, β

be the roots of

x^{2} - x - 1 = 0

and

S_{n} = α^{n} + β^{n}

, for all integers

n \geq 1

. Then for every integer

n \geq 2

Q. Let p

\geq

3 be an integer and

α

β

be the roots of

x^{2}

- (p + 1)x + 1 = 0.Using mathematical induction show that

α^{n} + β^{n}

is an integer

Q. Let p

\geq

3 be an integer and

α

β

be the roots of

x^{2}

- (p + 1)x + 1 = 0.Using mathematical induction show that

α^{n} + β^{n}

is not divisible by p.

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 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}}

Join BYJU'S Learning Program

Let α and β be the roots of x2+x+1=0. If n be positive integer, then αn+βn is

The correct option is A 2cos2nπ3x2+x+1=0⇒x=−1±√−32 Let α=−1+√3i2,β=−1−√3i2 ⇒α=exp(2πi3),β=exp(−2πi3) αn+βn=exp(2nπi3),β=exp(−2nπi3) =2cos2nπ3 [∵eiθ=cosθ+isinθ]

Let $α$ and $β$ be the roots of $x^{2} + x + 1 = 0$ . If $n$ be positive integer, then $α^{n} + β^{n}$ is