Let's work through the problem step by step.

### Step 1: Understanding the Pattern

You have a sequence of figures made of red and black squares. Let's denote:
- \( n \): The position number in the sequence (i.e., \( n = 1 \) for the first figure, \( n = 2 \) for the second, etc.).
- \( E_n \): The total number of squares in the figure at position \( n \).

### Step 2: Completing the Table

From the provided image, it seems like the number of red squares remains the same, while the number of black squares increases as \( n \) increases. Let's count the number of squares in each pattern and complete the table.

1. **Pattern 1 (n = 1):**
   - Number of Red Squares: 4
   - Number of Black Squares: 1
   - Total Number of Squares, \( E_1 \): 5

2. **Pattern 2 (n = 2):**
   - Number of Red Squares: 4
   - Number of Black Squares: 3
   - Total Number of Squares, \( E_2 \): 7

3. **Pattern 3 (n = 3):**
   - Number of Red Squares: 4
   - Number of Black Squares: 5
   - Total Number of Squares, \( E_3 \): 9

From this, we see a pattern in the total number of squares:
- \( E_1 = 5 \)
- \( E_2 = 7 \)
- \( E_3 = 9 \)

### Step 3: Identify the Sequence and Write the Formula

From the table, we can observe that:

- The sequence of total squares \( E_n \) increases by 2 each time: \( 5, 7, 9, \dots \)

This suggests a linear relationship between \( E_n \) and \( n \).

Let's hypothesize that the relationship is of the form:
\[ E_n = 2n + C \]
where \( C \) is a constant.

Using \( n = 1 \), we have \( E_1 = 5 \):
\[ 5 = 2(1) + C \]
\[ C = 3 \]

Thus, the general formula for the number of squares in each pattern is:
\[ E_n = 2n + 3 \]

### Step 4: Solve Specific Questions

1. **1.1.2 (a):** Length of the number pattern \( n \) is the sequence of natural numbers, which is infinite, so the length depends on the number of terms you choose to consider.
  
2. **1.1.2 (b):** The difference between successive terms of \( E_n \) is constant and equal to 2.

3. **1.1.2 (c):** The rule for the sequence \( E_n \) is \( E_n = 2n + 3 \).

4. **1.1.2 (d):** The formula \( E_n = 2n + 3 \) expresses \( E_n \) in terms of \( n \).

### Step 5: Application

**1.1.4 (a):** For \( n = 10 \), calculate \( E_{10} \):
\[ E_{10} = 2(10) + 3 = 20 + 3 = 23 \]

**1.1.5:** Interpreting the sum and adding the difference methodically would involve summing all \( E_n \) for the required range of \( n \).

Would you like more details on any of these steps or to discuss additional questions related to this problem?

Explore how to solve arithmetic sequence problems step-by-step. Learn about patterns in sequences, formulas for arithmetic sequences, and how to apply linear equations in sequence analysis. Suitable for Grades 6-8.

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