https://file.aiquickdraw.com/mathbot/file/1726161060194fapl6wb2.jpg

The exercise shown in the image is about the sum of terms in a sequence ("Somme des termes d'une suite").

### Problem Setup:
- The sequence \( (u_n) \) is defined by:
  - \( u_0 = -0.3 \)
  - For all integers \( n \geq 0 \), \( u_{n+1} = -2u_n \).
  
The recursive relationship can be expanded as:
\[ u_n = -0.3 \times (-2)^n \]

### Objective:
- Calculate the sum of the terms in the sequence up to \( n \), denoted as:
\[
S = \sum_{k=0}^{n} u_k = u_0 + u_1 + \dots + u_n
\]

Do you want me to proceed with solving this problem, calculating the sum explicitly, or explaining the formula step by step?

Let me know if you'd like more details!

Here are some questions that could extend this:

1. How do you find the explicit formula for a geometric series?
2. What is the general method to find the sum of a sequence?
3. How does recursion help in defining sequences?
4. Can we deduce the long-term behavior of this sequence?
5. What are the key differences between arithmetic and geometric sequences?

**Tip:** For a geometric series, you can often find a closed-form formula for the sum, which simplifies your calculations tremendously.

Calculate the sum of the terms of the sequence (u_n) defined by u_0 = -0.3 and for all integers n >= 0, u_{n+1} = -2u_n.

Learn how to calculate the sum of a geometric sequence defined by recursion. The problem explores the recursive relationship u_{n+1} = -2u_n, and how to find the sum of terms up to a given value of n. Suitable for high school students studying sequences and series.

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