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The problem in the image asks to prove an inequality involving sums of real strictly positive numbers \( a_1, a_2, \dots, a_n \), where \( n \in \mathbb{N} \). The expression to prove is:

\[
\left( \sum_{k=1}^{n} a_k \right) \left( \sum_{k=1}^{n} \frac{1}{a_k} \right) \geq n^2
\]

This inequality is a form of the **Cauchy-Schwarz inequality** applied in a specific context. To prove it, we will utilize the **arithmetic-harmonic mean inequality**.

### Proof Outline:
We will use the inequality between the arithmetic mean (AM) and harmonic mean (HM), which states that for positive real numbers \( a_1, a_2, \dots, a_n \), we have:

\[
\frac{a_1 + a_2 + \dots + a_n}{n} \geq \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}}
\]

Multiplying both sides by \( n \left( \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} \right) \), we get:

\[
\left( a_1 + a_2 + \dots + a_n \right) \left( \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} \right) \geq n^2
\]

This proves the desired inequality.

Would you like further details or any clarification on this proof?

Here are five related questions to deepen your understanding:
1. How does the Cauchy-Schwarz inequality apply in different contexts?
2. What is the relationship between the arithmetic and harmonic means?
3. Can you provide an example where this inequality is strict?
4. How does this inequality extend to other types of means (geometric or quadratic)?
5. How would you prove the Cauchy-Schwarz inequality directly in this case?

**Tip:** The arithmetic-harmonic mean inequality is a powerful tool when working with sums and reciprocals, especially in proving inequalities like this one.

Soient n ∈ N et a_1,...,a_n, réels strictement positifs. Montrer que (∑_{k=1}^{n} a_k)(∑_{k=1}^{n} (1/a_k)) ≥ n²

Explore the proof of the inequality (∑ a_k)(∑ 1/a_k) ≥ n², involving positive real numbers and applying the Arithmetic-Harmonic Mean inequality. This problem uses the Cauchy-Schwarz inequality and provides a step-by-step outline for understanding the solution.

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