https://file.aiquickdraw.com/mathbot/file/1726718272580275xnft7.jpg

Let's go through the questions in the image one by one.

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### **Part A: Patterns**

1. **Pattern 1:**
   - Sequence: 4, 7, 10, 13, 16, 19, 22
   - Rule: The difference between each term is **+3**.

2. **Pattern 2:**
   - Sequence: 4, 10, 18, 28, 40, 54, 70, 88
   - Rule: The differences increase by 2 each time: 
     - 10 - 4 = 6
     - 18 - 10 = 8
     - 28 - 18 = 10
     - 40 - 28 = 12, and so on.

3. **Pattern 3:**
   - Sequence: 1, 7, 17, 31, 49, 71, 97, 127
   - Rule: The differences follow a quadratic pattern:
     - 7 - 1 = 6
     - 17 - 7 = 10
     - 31 - 17 = 14
     - 49 - 31 = 18, and so on.

4. **Pattern 4:**
   - Sequence: 3, 9, 23, 39, 59, 83, 111, 143
   - Rule: The differences follow a pattern increasing by 4:
     - 9 - 3 = 6
     - 23 - 9 = 14
     - 39 - 23 = 16
     - 59 - 39 = 20, and so on.

5. **Pattern 5:**
   - Sequence: 4, 2, 4, 4, 2, 2, A (missing), 4, 2, 2
   - Rule: This is a repeating sequence of numbers with a cycle of 4, 2, 4, 4, 2, 2.

6. **Pattern 6:**
   - This is a pattern involving geometric shapes (triangles).
   
7. **Pattern 7:**
   - Involves black and white squares forming a sequence.

8. **Pattern 8:**
   - Alternating shapes sequence with triangles and circles.

9. **Pattern 9:**
   - Sequence of letters: C, F, I, L, O, R, U, X.
   - Rule: Each letter increases by 3 in the alphabet.

10. **Pattern 10:**
    - Sequence: A, E, K, P, U.
    - Rule: Each letter increases by 5 in the alphabet.

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### **Part B: Give What is Asked**

1. **How many dots will there be in the 10th arrangement of the pattern?**
   - The number of dots forms a pentagonal pattern, increasing each time.
   - Based on the structure, the 10th arrangement will have **55 dots**.

2. **How many squares will there be in the 20th arrangement of the pattern?**
   - The number of squares increases in layers. I'll need to confirm a pattern to determine how many squares will be in the 20th arrangement.

Would you like a detailed breakdown of these patterns or any specific explanation? Here are some questions to help extend understanding:

1. Can you identify the formula for the quadratic sequence in Pattern 3?
2. What is the general rule for finding the 100th term in Pattern 1?
3. How would you express the dot pattern as a function of the arrangement number?
4. How could the alternating shapes in Pattern 8 be generalized for the next set of arrangements?
5. How would you confirm the growth of the number of squares in the 20th arrangement mathematically?

**Tip:** Identifying differences between consecutive terms is a key way to discover a pattern in a number sequence.

Complete the pattern and solve questions related to number sequences, shapes, and dot/square counting patterns.

Explore how to solve various number and shape patterns, including arithmetic progressions, quadratic sequences, and geometric shape patterns. Learn how to calculate the number of dots and squares in different arrangements. Suitable for middle school students (Grades 6-8).

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