Question

If ∝ , are the roots of the equation x2

– 2x + 4 = 0 then for any n N, show that



+

= 2n+1. Cos (



/3

).​

Answer 1

Step-by-step explanation:

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Asked on December 30, 2019 by

Gouri Affan

If α,β are the roots of x

2

−2x+4=0, then α

5

+β

5

=

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Study later

ANSWER

α,β are the roots of x

2

−2x+4=0

⇒α+β=2,αβ=4

α

2

+β

2

=(α+β)

2

−2αβ

=2

2

−2(4)

=4−8=−4

α

3

+β

3

=(α+β)

3

−3αβ(α+β)

=2

3

−3(4)(2)

=8−24=−16

Now, α

5

+β

5

=(α

3

+β

3

)(α

2

+β

2

)−α

3

β

2

−α

2

β

3

=(α

3

+β

3

)(α

2

+β

2

)−(αβ)

2

(α+β)

=(−16)(−4)−(4)

2

(2)

=64−32=32