The average of seven consecutive odd numbers (in increasing order) is k. If the next four consecutive odd numbers are included, then the average of all the numbers is :

Options

A. k+7
B. k+3
C. k+4
D. k+2

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the seven consecutive odd numbers
Let the seven consecutive odd numbers be represented as:
- \( x - 6 \)
- \( x - 4 \)
- \( x - 2 \)
- \( x \)
- \( x + 2 \)
- \( x + 4 \)
- \( x + 6 \)

### Step 2: Calculate the average of these seven numbers
The average \( k \) of these seven numbers can be calculated as follows:
\[ k = \frac{(x - 6) + (x - 4) + (x - 2) + x + (x + 2) + (x + 4) + (x + 6)}{7} \]
Simplifying the numerator:
\[ = \frac{7x}{7} = x \]
Thus, we have \( k = x \).

### Step 3: Define the next four consecutive odd numbers
The next four consecutive odd numbers after the seven are:
- \( x + 8 \)
- \( x + 10 \)
- \( x + 12 \)
- \( x + 14 \)

### Step 4: Calculate the total of all eleven numbers
Now, we will find the total of all eleven numbers:
\[ \text{Total} = (x - 6) + (x - 4) + (x - 2) + x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) + (x + 12) + (x + 14) \]
This simplifies to:
\[ = 7x + (8 + 10 + 12 + 14) = 7x + 44 \]

### Step 5: Calculate the average of all eleven numbers
Now, we calculate the average of these eleven numbers:
\[ \text{Average} = \frac{7x + 44}{11} \]
Substituting \( k \) for \( x \):
\[ = \frac{7k + 44}{11} \]

### Step 6: Simplify the average
To express this in terms of \( k \):
\[ = \frac{7k}{11} + \frac{44}{11} = \frac{7k}{11} + 4 \]

### Conclusion
Thus, the average of all the numbers when the next four consecutive odd numbers are included is:
\[ \frac{7k + 44}{11} \]