https://file.aiquickdraw.com/mathbot/file/1734258726545j0bh8oj1.jpg

To determine the number of tiles in the **5th** and **6th** patterns, let's analyze the sequence.

### Observing the Rule
- The sequence starts with **1 tile** in the 1st pattern.
- The number of tiles **doubles** with each step.

This means the sequence follows a **geometric progression** with:
- First term \( a = 1 \)
- Common ratio \( r = 2 \)

The formula for the number of tiles in the \( n \)-th pattern is:

\[
T_n = a \cdot r^{n-1}
\]

Here:
- \( a = 1 \)
- \( r = 2 \)

### Calculating for the 5th Pattern (\( n = 5 \))
\[
T_5 = 1 \cdot 2^{5-1} = 2^4 = 16
\]

### Calculating for the 6th Pattern (\( n = 6 \))
\[
T_6 = 1 \cdot 2^{6-1} = 2^5 = 32
\]

### Final Answer:
- **5th pattern**: 16 tiles  
- **6th pattern**: 32 tiles  

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Do you want further explanation or details on geometric sequences? Here are 5 follow-up questions to consider:  
1. What is a geometric progression, and how is it identified?  
2. How can you derive the general formula for geometric sequences?  
3. What happens if the doubling rule changes to tripling?  
4. Can you graph the number of tiles for the first 10 patterns?  
5. How would the sequence change if the starting number of tiles was 2 instead of 1?  

**Tip**: In geometric sequences, each term is obtained by multiplying the previous term by a constant ratio.

Mandy arranges tiles to form a sequence of patterns. Work out how many tiles there are in each of the 5th and 6th patterns.

Learn how to calculate the number of tiles in the 5th and 6th patterns of a doubling sequence. This problem introduces geometric progression concepts, exponential growth, and practical problem-solving for Grades 5-7 students.

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