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From the uploaded image, I can see a table of values and some instructions. The task involves determining the missing values in the columns titled \(m\) and \(w(m)\), based on the given \(k\) values. To guide you through the solution, let's break it down step by step:

1. **Identifying the pattern**:
   We are given:
   \[
   \begin{array}{|c|c|c|}
   \hline
   k & m & w(m) \\
   \hline
   1 & 12 & 0 \\
   3 & 16 & 10 \\
   5 & 19 & 18 \\
   7 & ? & 25 \\
   \hline
   \end{array}
   \]

2. **Finding a pattern for \(m\)**:
   - The values for \(m\) in the table appear to be increasing as \(k\) increases.
   - Let's observe how \(m\) changes: 
     - From \(m = 12\) to \(m = 16\), the change is \(+4\).
     - From \(m = 16\) to \(m = 19\), the change is \(+3\).

     Following this pattern, the change should alternate between increasing by 4 and 3. So for \(k = 7\), the change should be \(+4\). Thus:
     \[
     m = 19 + 4 = 23
     \]

3. **Finding a pattern for \(w(m)\)**:
   - The values for \(w(m)\) also follow a pattern:
     - From \(w(m) = 0\) to \(w(m) = 10\), the change is \(+10\).
     - From \(w(m) = 10\) to \(w(m) = 18\), the change is \(+8\).
     - From \(w(m) = 18\) to \(w(m) = 25\), the change is \(+7\).

     It seems like the changes are decreasing by 1 each time. So the next change should be \(+6\). Thus, for \(k = 7\):
     \[
     w(m) = 25 + 6 = 31
     \]

4. **Final table**:
   After filling in the missing values, the table should look like:
   \[
   \begin{array}{|c|c|c|}
   \hline
   k & m & w(m) \\
   \hline
   1 & 12 & 0 \\
   3 & 16 & 10 \\
   5 & 19 & 18 \\
   7 & 23 & 31 \\
   \hline
   \end{array}
   \]

Would you like further explanation on any of these steps?

Here are 5 related questions to deepen your understanding:
1. How could you verify the pattern in \(m\) values if it were more complex?
2. What could you infer about \(w(m)\) if the change in values were irregular?
3. How would you generalize the approach to solve similar problems with different sequences?
4. Can you find an explicit formula for \(w(m)\) in terms of \(k\)?
5. How would the solution change if there was a different alternating pattern for \(m\)?

**Tip**: Always check if there is an alternating pattern in a sequence of numbers when the changes are not constant!

Complete the table and find the missing values for m and w(m) based on the provided values of k.

Learn how to solve a problem by recognizing patterns in sequences. The problem involves finding missing values in the sequences of m and w(m) based on changes in the variable k. Suitable for Grades 7-9 students.

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