Let `z_(1),z_(2),z_(3),……z_(n)` be the complex numbers such that `|z_(1)|= |z_(2)| = …..=|z_(n)| = 1`. If `z = (sum_(k=1)^(n)z_(k)) (sum_(k=1)^(n)(1)/(z_(k)))` then prove that : z is a real number .

Explanation

To prove that \( z \) is a real number given that \( |z_1| = |z_2| = \ldots = |z_n| = 1 \), we start with the expression:

\[ z = \left( \sum_{k=1}^{n} z_k \right) \left( \sum_{k=1}^{n} \frac{1}{z_k} \right) \]

### Step 1: Rewrite the expression for \( z \)

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