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?