If `omega` is the `2014^("th")` root of unity and `z_(1)` and `z_(2)` be any two complex numbers such that `|z_(1)|=3,|z_(2)|=4` then value of `underset(k=0)overset(2013)sum |z_(1)+omega^(k)z_(2)|^(2)` is

Options

A. 50250
B. 50350
C. 50450
D. 50550

Answer

To solve the problem, we need to evaluate the expression:

\[ \sum_{k=0}^{2013} |z_1 + \omega^k z_2|^2 \]

where \(\omega\) is the \(2014^{th}\) root of unity, and \(|z_1| = 3\) and \(|z_2| = 4\).

### Step 1: Expand the Expression

We start by expanding the expression inside the summation:

\[ |z_1 + \omega^k z_2|^2 = (z_1 + \omega^k z_2)(\overline{z_1 + \omega^k z_2}) = (z_1 + \omega^k z_2)(\overline{z_1} + \overline{\omega^k z_2}) \]

Using the properties of complex conjugates, we have \(\overline{\omega^k} = \omega^{-k}\) since \(\omega\) is a root of unity.

Thus, we can rewrite it as:

\[ |z_1 + \omega^k z_2|^2 = |z_1|^2 + |z_2|^2 + z_1 \overline{\omega^k z_2} + \overline{z_1} \omega^k z_2 \]

### Step 2: Substitute the Magnitudes

Given \(|z_1| = 3\) and \(|z_2| = 4\), we have:

\[ |z_1|^2 = 9 \quad \text{and} \quad |z_2|^2 = 16 \]

So, the expression becomes:

\[ |z_1 + \omega^k z_2|^2 = 9 + 16 + z_1 \overline{\omega^k z_2} + \overline{z_1} \omega^k z_2 \]

### Step 3: Sum Over k

Now, we need to sum this expression from \(k=0\) to \(k=2013\):

\[ \sum_{k=0}^{2013} |z_1 + \omega^k z_2|^2 = \sum_{k=0}^{2013} (9 + 16) + \sum_{k=0}^{2013} (z_1 \overline{\omega^k z_2} + \overline{z_1} \omega^k z_2) \]

The first part simplifies to:

\[ \sum_{k=0}^{2013} (9 + 16) = 2014 \times 25 = 50350 \]

### Step 4: Evaluate the Second Part

Next, we need to evaluate:

\[ \sum_{k=0}^{2013} (z_1 \overline{\omega^k z_2} + \overline{z_1} \omega^k z_2) \]

Notice that \(\sum_{k=0}^{2013} \omega^k\) is the sum of a geometric series. Since \(\omega\) is a \(2014^{th}\) root of unity, this sum equals zero:

\[ \sum_{k=0}^{2013} \omega^k = 0 \]

Thus, both terms \(z_1 \overline{z_2} \sum_{k=0}^{2013} \omega^k\) and \(\overline{z_1} z_2 \sum_{k=0}^{2013} \omega^k\) are zero.

### Final Result

Therefore, the entire sum simplifies to:

\[ \sum_{k=0}^{2013} |z_1 + \omega^k z_2|^2 = 50350 + 0 = 50350 \]

Thus, the final answer is:

\[ \boxed{50350} \]