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The image contains a math problem related to geometric sequences. Here's a transcription and translation of the problem:

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**Exercise 2: Geometric Sequences**

We consider the sequence \((u_n)\), defined by \(u_1 = \frac{1}{2}\) and the recurrence relation \(u_{n+1} = \frac{u_n}{1 + 2u_n}\) for all \(n \geq 1\).

1. Show that \(u_n > 0\) for any natural number \(n\).
2. The sequence \((u_n)\) is decreasing; justify this.
3. Determine the limit of \((u_n)\) as \(n\) approaches infinity.

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We consider the sequence (u_n), defined by u_1 = 1/2 and the recurrence relation u_(n+1) = u_n / (1 + 2u_n) for all n >= 1. 1. Show that u_n > 0 for any natural number n. 2. The sequence (u_n) is decreasing; justify this. 3. Determine the limit of (u_n) as n approaches infinity.

Explore the geometric sequence defined by u_1 = 1/2 and the recurrence relation u_(n+1) = u_n / (1 + 2u_n). Learn how to prove that u_n > 0 for all natural numbers, demonstrate that the sequence is decreasing, and determine the limit as n approaches infinity. Suitable for students in Grades 10-12.

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