https://file.aiquickdraw.com/mathbot/file/17260922470428pvdv80w.jpg

The image you uploaded includes two parts of a mathematical exercise, labeled as *Partie A* and *Partie B*. Let me break down the content:

### Partie A:
This section deals with a geometric sequence \((u_n)\), with the following details:

1. \( u_n = 25000 \times 0.94^n \), for every natural number \( n \).
2. **Questions:**
   - (1) Determine an explicit formula for the sequence \((u_n)\).
   - (2) Calculate the sum of the first five terms of the sequence \((u_n)\).
   - (3) Compare the terms \( u_0 \) and \( u_1 \), then \( u_0 \) and \( u_3 \).
   - (4) Find the smallest integer \( n \) such that \( u_n < u_1 \).

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### Partie B:
This part relates to a car dealership's sales data for cars with gasoline and diesel engines:

- In 1995, the dealership sold **5200 gasoline cars**.
- The sales of gasoline cars **increase by 1250 units each year**, while the sales of **diesel cars decrease by 6% each year**. 
- In the first year, the dealership sold **25,000 diesel cars**.
  
**Question:**
- In what year will the sales of gasoline cars exceed those of diesel cars?

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Would you like me to solve both parts or focus on a specific question? Let me know if you'd like more detailed explanations or clarifications for any part.

### Related Questions:
1. What are the key properties of geometric sequences that help in solving such problems?
2. How can you determine the sum of a geometric sequence efficiently?
3. Why is the comparison of terms important when analyzing a sequence?
4. How can we model real-world scenarios like car sales with sequences or series?
5. What are the methods to solve for when one quantity surpasses another in real-world models?

#### Tip:
In geometric sequences, the ratio between consecutive terms remains constant, which makes predicting long-term behavior (growth or decay) very useful for modeling situations like sales decline.

A mathematical exercise involving a geometric sequence and car sales, with two parts. In Partie A, a sequence is defined as u_n = 25000 × 0.94^n, and the task includes finding the explicit formula, calculating the sum of the first five terms, comparing terms, and determining the smallest n such that u_n < u_1. In Partie B, car sales data is given, where gasoline car sales increase by 1250 units yearly, and diesel car sales decrease by 6% yearly. The task is to determine when gasoline car sales will exceed diesel car sales.

Solve a problem involving geometric sequences and real-world car sales data. Calculate sums, compare terms, and determine when gasoline car sales will surpass diesel car sales. Learn key concepts like exponential decay and arithmetic progression, perfect for Grades 10-12.

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