If {z_1}, {z_2}, {z_3}, ......, {z_n} are the | vedclass

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(D) The $n^{th}$ roots of unity are given by the formula: ${z_k} = {e^{\frac{i 2\pi (k - 1)}{n}}}$ for $k = 1, 2, ....., n$. Taking the modulus of bot

Vedclass · Accepted Answer

$|{z_k}| = |{z_{k + 1}}|$

Vedclass · Answer

(D) The $n^{th}$ roots of unity are given by the formula: ${z_k} = {e^{\frac{i 2\pi (k - 1)}{n}}}$ for $k = 1, 2, ....., n$. Taking the modulus of both sides: $|{z_k}| = |{e^{\frac{i 2\pi (k - 1)}{n}}}| = 1$ for all $k = 1, 2, ....., n$. Since the modulus of every $n^{th}$ root of unity is $1$,it follows that: $|{z_k}| = |{z_{k + 1}}| = 1$ for all $k = 1, 2, ....., n-1$. Therefore,option $D$ is correct.

If ${z_1}, {z_2}, {z_3}, ......, {z_n}$ are the $n^{th}$ roots of unity,then for $k = 1, 2, ....., n-1$:

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