Question

If

w \neq 1

is

n^{th}

root of unity, then find the value of

\sum_{k = 0}^{n - 1} {|z_{1} + w^{k} z_{2}|}^{2}

.

Accepted Answer

We have to find the value of $\sum_{k = 0}^{n - 1} ∣ z_{1} + w^{k} z_{2} l^{2}$

For a complex number $z$ , we know that, ${|z|}^{2} = z \bar{z}$ , and $\bar{z_{i} + z_{j}} = \bar{z_{i}} + \bar{z_{j}}$ ,

Using the properties mentioned above $\sum_{k = 0}^{n - 1} ∣ z_{1} + w^{k} z_{2} l^{2}$ $= \sum_{k = 0}^{n - 1} |z_{1} + w^{k} z_{2}| ∣ z_{1} + w^{k} z_{2} | = \sum_{k = 0}^{n - 1} (z_{1} + w^{k} z_{2}) (\bar{z_{1}} + \bar{w^{k} z_{2}}) = \sum_{k = 0}^{n - 1} (z_{1} + w^{k} z_{2}) (\bar{z_{1}} + \bar{w^{k}} \bar{z_{2}})$

$\sum_{k = 0}^{n - 1} [z_{1} \bar{z_{1}} + w^{k} z_{2} \bar{z_{1}} + \bar{w^{k}} \bar{z_{2}} z_{1} + w^{k} z_{2} \bar{w^{k}} \bar{z_{2}}] = \sum_{k = 0}^{n - 1} z_{1} \bar{z_{1}} + \sum_{k = 0}^{n - 1} w^{k} z_{2} \bar{z_{1}} + \sum_{k = 0}^{n - 1} \bar{w^{k}} \bar{z_{2}} z_{1} + \sum_{k = 0}^{n - 1} w^{k} z_{2} \bar{w^{k}} \bar{z_{2}}$

$= \sum_{k = 0}^{n - 1} {|z_{1}|}^{2} + z_{2} \bar{z_{1}} \sum_{k = 0}^{n - 1} w^{k} + \bar{z_{2}} z \sum_{k = 0}^{n - 1} {\bar{w^{k}}}_{1} + z_{2} \bar{z_{2}} \sum_{k = 0}^{n - 1} w^{k} \bar{w^{k}} = \sum_{k = 0}^{n - 1} {|z_{1}|}^{2} + z_{2} \bar{z_{1}} \sum_{k = 0}^{n - 1} w^{k} + \bar{z_{2}} z_{1} \sum_{k = 0}^{n - 1} \bar{w^{k}} + z_{2} \bar{z_{2}} \sum_{k = 0}^{n - 1} {({|w|}^{2})}^{k}$
As, $w$ is $nth$ root of unity,

$\sum_{k = 0}^{n - 1} w^{k} = \sum_{k = 0}^{n - 1} \bar{w^{k}} = 0$ and ${|w|}^{2} = 1$

Now, substituting these values we get,

$\sum_{k = 0}^{n - 1} {|z_{1}|}^{2} + z_{2} \bar{z_{1}} \sum_{k = 0}^{n - 1} w^{k} + \bar{z_{2}} z_{1} \sum_{k = 0}^{n - 1} \bar{w^{k}} + z_{2} \bar{z_{2}} \sum_{k = 0}^{n - 1} ({|w|}^{2}) = n {|z_{1}|}^{2} + z_{2} \bar{z_{1}} (0) + \bar{z_{2}} z_{1} (0) + z_{2} \bar{z_{2}} (n)$

$= n ({|z_{1}|}^{2} + {|z_{2}|}^{2})$

Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 4: Exercise-4

Embibe Experts Mathematics Solutions for Exercise - Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 4: Exercise-4

Questions from Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 4: Exercise-4 with Hints & Solutions

Learn and Prepare for any exam you want !