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Let's analyze and solve the questions related to the image of geometric sequences.

### Q1: 
The given sequence is \( 50, 10, 2, \ldots \). We need to find the next three terms.
1. **Step 1:** Find the common ratio \( r \) of the sequence.
   - To find the common ratio, divide the second term by the first term:
     \[
     r = \frac{10}{50} = 0.2
     \]
2. **Step 2:** Apply the common ratio to find the next terms:
   - Fourth term: \( 2 \times 0.2 = 0.4 \)
   - Fifth term: \( 0.4 \times 0.2 = 0.08 \)
   - Sixth term: \( 0.08 \times 0.2 = 0.016 \)

Thus, the next three terms are **0.4, 0.08, and 0.016**.

### Q2: 
1. **What is the common ratio \( r \)?**
   - From the previous calculation, the common ratio \( r = 0.2 \).

2. **What is the formula for the \( n \)-th term?**
   The general formula for the \( n \)-th term of a geometric sequence is:
   \[
   T_n = T_1 \times r^{n-1}
   \]
   For this sequence:
   \[
   T_n = 50 \times (0.2)^{n-1}
   \]

### Q3: Work out the first three terms of the sequences with the given formulas.
1. **For the sequence \( 5 \times \sqrt{5}^{n-1} \):**
   - First term: \( 5 \times (\sqrt{5})^0 = 5 \times 1 = 5 \)
   - Second term: \( 5 \times (\sqrt{5})^1 = 5 \times \sqrt{5} \)
   - Third term: \( 5 \times (\sqrt{5})^2 = 5 \times 5 = 25 \)

2. **For the sequence \( -1 \times (-5)^{n-1} \):**
   - First term: \( -1 \times (-5)^0 = -1 \times 1 = -1 \)
   - Second term: \( -1 \times (-5)^1 = -1 \times (-5) = 5 \)
   - Third term: \( -1 \times (-5)^2 = -1 \times 25 = -25 \)

### Final Question:
- **The second term of a geometric sequence is 10.5 and the third term is 73.5.**
  - We know that \( T_2 = 10.5 \) and \( T_3 = 73.5 \). Using the formula for the \( n \)-th term:
    \[
    T_2 = T_1 \times r = 10.5
    \]
    \[
    T_3 = T_1 \times r^2 = 73.5
    \]
    Divide \( T_3 \) by \( T_2 \) to find the common ratio:
    \[
    \frac{T_3}{T_2} = r = \frac{73.5}{10.5} = 7
    \]
    So, \( r = 7 \). Now use this to find \( T_1 \):
    \[
    T_2 = T_1 \times 7 \quad \Rightarrow \quad 10.5 = T_1 \times 7 \quad \Rightarrow \quad T_1 = \frac{10.5}{7} = 1.5
    \]

Thus, the first term is **1.5** and the common ratio is **7**.

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Would you like more details or have any questions on this? Here are some additional related questions:

1. How do you derive the formula for the \( n \)-th term of a geometric sequence?
2. Can you find the sum of the first 10 terms of the sequence \( 50, 10, 2, \ldots \)?
3. How does the common ratio affect the behavior of a geometric sequence (e.g., increasing, decreasing)?
4. Can you explore geometric sequences with negative common ratios?
5. What happens if the common ratio is greater than 1 or less than 1?

**Tip:** In geometric sequences, if \( |r| < 1 \), the terms approach 0 as \( n \) increases!

Find the next three terms of the sequence 50, 10, 2, ... and answer questions related to geometric sequences.

Learn how to solve geometric sequences by finding the next three terms of the sequence 50, 10, 2, ... and understand the nth term formula. Includes detailed solutions for finding the common ratio and applying formulas. Suitable for students in Grades 9-11.

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