If-z-is-a-complex-number-satisfying-z-z-1-1-then-z-n-z-n-n-N-has-the-value-1-2-1-n-when-n-is-a-multiple-of-3-2-1-n-1-when-n-is-not-a-multiple-of-3-3-1-n-1-w

Question Number 20933 by Tinkutara last updated on 08/Sep/17

If z is a complex number satisfying

z + z^{- 1} = 1, then z^{n} + z^{- n}, n \in N, has

the value

(1) 2 {(- 1)}^{n}, when n is a multiple of 3

(2) {(- 1)}^{n - 1}, when n is not a multiple of

3

(3) {(- 1)}^{n + 1}, when n is a multiple of 3

(4) 0 when n is not a multiple of 3

Answered by $@ty@m last updated on 09/Sep/17

G i v e n

z + \frac{1}{z} = 1

\Rightarrow z^{2} - z + 1 = 0

\Rightarrow z = \frac{1 \pm i \sqrt{3}}{2}

\Rightarrow z = \cos \frac{π}{3} \pm i \sin \frac{π}{3}

∴ z^{n} + z^{- n} = \cos \frac{n π}{3} + i \sin \frac{n π}{3} + \cos \frac{n π}{3} - i \sin \frac{n π}{3}

= 2 c o s \frac{n π}{3}

Case (i) n = 1

2 c o s \frac{n π}{3} = 2 \cos \frac{π}{3} = 2 \times \frac{1}{2} = 1

Case (ii) n = 2

2 c o s \frac{n π}{3} = 2 \cos \frac{2 π}{3} = 2 \times \frac{- 1}{2} = - 1

Case (iii) n = 3

2 c o s \frac{n π}{3} = 2 \cos π = 2 \times (- 1) = - 2

a n d s o o n \dots .

\Rightarrow (1) & (2) a r e c o r r e c t a n s w e r s .

Commented by Tinkutara last updated on 09/Sep/17

Thank you very much Sir!