The average of seven consectutive 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 step by step, we will follow the logic of the video transcript while providing a clear explanation.

### Step 1: Define the seven consecutive odd numbers
Let the first of the seven consecutive odd numbers be represented as \( A \). Therefore, the seven consecutive odd numbers can be expressed as:
- \( A \)
- \( A + 2 \)
- \( A + 4 \)
- \( A + 6 \)
- \( A + 8 \)
- \( A + 10 \)
- \( A + 12 \)

### Step 2: Calculate the average of the seven numbers
The sum of these seven numbers is:
\[ A + (A + 2) + (A + 4) + (A + 6) + (A + 8) + (A + 10) + (A + 12) = 7A + 42 \]
The average \( k \) of these seven numbers is given by:
\[ k = \frac{7A + 42}{7} \]
Simplifying this gives:
\[ k = A + 6 \]

### Step 3: Express \( A \) in terms of \( k \)
From the equation \( k = A + 6 \), we can express \( A \) as:
\[ A = k - 6 \]

### Step 4: Define the next four consecutive odd numbers
The next four consecutive odd numbers after the first seven are:
- \( A + 14 \)
- \( A + 16 \)
- \( A + 18 \)
- \( A + 20 \)

### Step 5: Calculate the sum of all eleven numbers
The total sum of all eleven numbers (the first seven plus the next four) is:
\[ (7A + 42) + (A + 14 + A + 16 + A + 18 + A + 20) = 7A + 42 + 4A + 68 = 11A + 110 \]

### Step 6: Calculate the new average
The new average of all eleven numbers is:
\[ \text{New Average} = \frac{11A + 110}{11} \]
Simplifying this gives:
\[ \text{New Average} = A + 10 \]

### Step 7: Substitute \( A \) back in terms of \( k \)
Substituting \( A = k - 6 \) into the new average:
\[ \text{New Average} = (k - 6) + 10 = k + 4 \]

### Conclusion
Thus, the average of all the numbers when the next four consecutive odd numbers are included is:
\[ \text{New Average} = k + 4 \]