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Question Number 21307 by Tinkutara last updated on 20/Sep/17

Let z_{1}, z_{2}, z_{3} be complex numbers such

that

(i) ∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = 1

(ii) z_{1} + z_{2} + z_{3} \neq 0

(iii) z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 0

Prove that for all n ⩾ 2,

∣ z_{1}^{n} + z_{2}^{n} + z_{3}^{n} ∣ \in {0, 1, 2, 3} .

Commented by Tinkutara last updated on 25/Sep/17

help pls

Commented by Tinkutara last updated on 24/Sep/17

help pls

Commented by Tinkutara last updated on 26/Sep/17

help pls

Commented by Tinkutara last updated on 27/Sep/17

L e t z_{k} = \cos θ_{k} + i \sin θ_{k}; k \in {1, 2, 3}

G i v e n t h a t z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 0

\cos 2 θ_{1} + \cos 2 θ_{2} = - \cos 2 θ_{3}

\sin 2 θ_{1} + \sin 2 θ_{2} = - \sin 2 θ_{3}

S q u a r i n g a n d a d d i n g,

2 + 2 \cos 2 (θ_{1} - θ_{2}) = 1

θ_{1} - θ_{2} = k π \pm \frac{π}{3}

S o l e t z_{1}^{n} = \cos n θ + i \sin n θ

z_{2}^{n} = \cos n (\frac{π}{3} + θ) + i \sin n (\frac{π}{3} + θ)

z_{3}^{n} = \cos n (\frac{2 π}{3} + θ) + i \sin n (\frac{2 π}{3} + θ)

∴∣ z_{1}^{n} + z_{2}^{n} + z_{3}^{n} ∣= \sqrt{3 + 2 \sum_{c y c} \cos n (θ_{1} - θ_{2})}

N o w I o n l y g o t 0 i n a l l c a s e s . C a n

s o m e o n e c o n t i n u e a f t e r t h i s ?

Commented by Tinkutara last updated on 22/Mar/18