Question

If a,ß are the roots of the equation x^2- 2x + 4 = 0 then for any positive integer n
show that a^n+B^n = 2^(n+1)cos(nπ÷3)​

Answer 1

Answer:

ANSWER

x

2

−2x+4=0

∴x=

2

2±

4−16

=1±

3

i

∴α=re

iθ

,β=re

−iθ

wheree=2andθ=π/3

∴α

n

+β

n

=r

n

(e

inθ

+e

−inθ

)

=r

n

[cosnθ+isinnθ+cosnθ−isinnθ]

=r

n

.2cosnθ=2

n+1

cos(nπ/3)