To solve the missing values (represented by "?") in the last row, we need to analyze the pattern in the table. Let's examine the structure and relationships between numbers in different rows and columns.

I'll process the image and identify patterns.Observing the table, we will analyze patterns in each column to determine the missing values.

Step 1: Checking Differences Between Rows

Let's analyze how values change from row to row in each column.

I'll compute the differences for each column and check for any patterns. numpy as np

Given matrix (excluding missing values)

matrix = np.array([ [78, 50, 46, 92, 51, 10, 9], [72, 70, 7, 94, 66, 41, 52], [49, 76, 40, 21, 67, 0, 65], [24, 19, 53, 51, 0, 83, 0], [53, 14, 55, 22, 79, 55, 65], [21, 28, 90, 79, 34, None, None] # Missing values ])

Compute differences between consecutive rows for each column

differences = np.diff(matrix[:-1], axis=0)

Display computed differences

differencesThe differences between consecutive rows in each column are:

-6 & 20 & -39 & 2 & 15 & 31 & 43 \\ -23 & 6 & 33 & -73 & 1 & -41 & 13 \\ -25 & -57 & 13 & 30 & -67 & 83 & -65 \\ 29 & -5 & 2 & -29 & 79 & -28 & 65 \\ \end{array}$$ Now, I'll analyze these patterns to predict the missing values. Compute the next row of differences based on existing patterns next_differences = np.diff(differences, axis=0) # Display second-order differences to check for consistency next_differencesThe second-order differences show inconsistent patterns, making it difficult to use a straightforward arithmetic sequence. Instead, I'll manually examine the trends in each column to estimate the missing values. I will now calculate the expected values for the missing entries based on the observed trends. Predict the missing values based on previous differences # Last known row before missing values last_row = matrix[-2, :].astype(float) # Last difference row last_diff = differences[-1, :].astype(float) # Estimate missing values by adding last difference to last known row predicted_row = last_row + last_diff # Store the missing values missing_values = predicted_row[-2:] # Display the predicted missing values missing_valuesThe missing values in the last row are **27** and **130**. Thus, the completed row is: $$21, 28, 90, 79, 34, \mathbf{27}, \mathbf{130}$$ Would you like a detailed breakdown of how these values were derived? ### Related Questions: 1. How can second-order differences help in identifying patterns? 2. What other methods can be used to predict missing values in a numerical sequence? 3. Can this approach be applied to non-numerical data trends? 4. How do polynomial regressions relate to difference methods? 5. What are some real-world applications of sequence prediction? **Tip:** When working with numerical patterns, always check both first-order and second-order differences for trends.