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The image provided contains a problem related to complex numbers. Here's a transcription of the problem and its instructions:

**Problem Statement:**
1. \(\alpha\) is a complex number such that \(\alpha + \frac{1}{\alpha}\) is a real number.
   
   If \(\alpha = r\left(\cos\theta + i\sin\theta\right)\), show that \(r = 1\).

   Hence, given that \(z = \alpha + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5\), determine \(z\).

---

**Solution:**

1. **Given**: \(\alpha = r(\cos\theta + i\sin\theta)\) and \(\alpha + \frac{1}{\alpha}\) is real.

   Start by calculating \(\frac{1}{\alpha}\):
   \[
   \frac{1}{\alpha} = \frac{1}{r(\cos\theta + i\sin\theta)} = \frac{1}{r}(\cos(-\theta) + i\sin(-\theta)) = \frac{1}{r}(\cos\theta - i\sin\theta)
   \]

   Therefore:
   \[
   \alpha + \frac{1}{\alpha} = r(\cos\theta + i\sin\theta) + \frac{1}{r}(\cos\theta - i\sin\theta)
   \]
   Simplifying, this becomes:
   \[
   \alpha + \frac{1}{\alpha} = \left(r + \frac{1}{r}\right)\cos\theta + i\left(r - \frac{1}{r}\right)\sin\theta
   \]
   Since \(\alpha + \frac{1}{\alpha}\) is real, the imaginary part must be zero. Therefore:
   \[
   \left(r - \frac{1}{r}\right)\sin\theta = 0
   \]
   For this to hold true, either \(\sin\theta = 0\) or \(r = 1\).

   - If \(\sin\theta = 0\), then \(\theta = n\pi\) where \(n\) is an integer. However, this doesn't provide any information about \(r\).
   - If \(r = 1\), the equation simplifies to:
   \[
   \alpha + \frac{1}{\alpha} = 2\cos\theta
   \]
   which is a real number. Therefore, \(r = 1\) is the required solution.

2. **Determining \(z\)**:

   Given \(z = \alpha + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5\) and \(r = 1\), this implies that \(\alpha = \cos\theta + i\sin\theta = e^{i\theta}\).

   Hence:
   \[
   z = e^{i\theta} + e^{2i\theta} + e^{3i\theta} + e^{4i\theta} + e^{5i\theta}
   \]
   This is a geometric series with the first term \(a = e^{i\theta}\) and common ratio \(r = e^{i\theta}\).

   The sum of the series is:
   \[
   z = \frac{e^{i\theta}(1 - (e^{i\theta})^5)}{1 - e^{i\theta}} = \frac{e^{i\theta}(1 - e^{5i\theta})}{1 - e^{i\theta}}
   \]
   Simplifying further:
   \[
   z = \frac{e^{i\theta}(1 - (\cos(5\theta) + i\sin(5\theta)))}{1 - (\cos\theta + i\sin\theta)}
   \]

   If \(\theta = \frac{2\pi}{5}\), this expression will give a specific value for \(z\). Without further constraints on \(\theta\), the exact value of \(z\) remains in this form.

Would you like further details or have any questions about this? Here are some related questions you might find interesting:

1. What is the geometric interpretation of complex numbers in the form \(r(\cos\theta + i\sin\theta)\)?
2. How can De Moivre's Theorem be used to simplify powers of complex numbers?
3. What are the roots of unity and how do they relate to this problem?
4. How can the argument of a complex number affect the sum of a series like the one in this problem?
5. What is the significance of the magnitude \(r = 1\) in complex number theory?

**Tip**: When working with complex numbers in polar form, always check if the magnitude (r) is 1, as this can often simplify calculations significantly.

Given that α is a complex number such that α + 1/α is a real number, show that r = 1, where α = r(cosθ + i sinθ). Then, given that z = α + α^2 + α^3 + α^4 + α^5, determine z.

Solve a complex number problem where α + 1/α is real, showing that r = 1 and determining z = α + α^2 + α^3 + α^4 + α^5. This advanced problem involves concepts like polar form, De Moivre's Theorem, and geometric series.

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