Is-i-0-i-1-i-2-i-n-cyclic-for-any-value-of-n-Determine-the-smallest-such-n-if-it-exists-is-a-complex-cuberoot-of-unity-and-i-1-

Question Number 8301 by Rasheed Soomro last updated on 06/Oct/16

Is {{(ω + i)}^{0}, {(ω + i)}^{1}, {(ω + i)}^{2}, \dots ., {(ω + i)}^{n}}

cyclic for any value of n ?

Determine the smallest such n if it exists .

ω is a complex cuberoot of unity and

i = \sqrt{- 1}

Commented by 123456 last updated on 07/Oct/16

{(ω + i)}^{n} = \underset{i = 0}{\sum^{n}} (\begin{matrix} n \\ i \end{matrix}) ω^{i} i^{n - i}

ω^{i} = ω^{i \mod 3}

i^{n - i} = i^{n - i \mod 4}

Answered by prakash jain last updated on 08/Oct/16

w = \cos \frac{2 π}{3} + i \sin \frac{2 π}{3}

w + i = \cos \frac{2 π}{3} + i (\sin \frac{2 π}{3} + 1)

∣ w + i ∣= \sqrt{\cos^{2} \frac{2 π}{3} + {(\sin \frac{2 π}{3} + 1)}^{2}} = r \neq 1

w + i = {r e}^{i \emptyset}

∵ r \neq 1, {(w + i)}^{j} = {(w + i)}^{k} \Rightarrow j = k