The average of seven consecutive odd numbers is A. If next four and previous three odd numbers to these seven odd numbers are also included, then what is the new average of these 14 consecutive odd numbers?

Options

A. A+3
B. A+4
C. A+2
D. A+1

Answer

To solve the problem, we need to find the new average of 14 consecutive odd numbers when we know the average of 7 consecutive odd numbers is A.

### Step-by-Step Solution:

1. **Understanding the Average of 7 Odd Numbers**:
Let the 7 consecutive odd numbers be represented as:
- \( n, n+2, n+4, n+6, n+8, n+10, n+12 \)
The average \( A \) of these numbers can be calculated as:
\[ A = \frac{n + (n+2) + (n+4) + (n+6) + (n+8) + (n+10) + (n+12)}{7} \]
Simplifying the numerator:
\[ A = \frac{7n + (0 + 2 + 4 + 6 + 8 + 10 + 12)}{7} = \frac{7n + 42}{7} = n + 6 \]
Thus, we can express \( n \) in terms of \( A \):
\[ n = A - 6 \]

2. **Identifying the Previous and Next Odd Numbers**:
The previous three odd numbers are:
- \( n-2, n, n+2 \)
- Specifically, they are \( (A-6)-2, (A-6), (A-6)+2 \) which gives us \( A-8, A-6, A-4 \).

The next four odd numbers are:
- \( n+14, n+12, n+10, n+8 \)
- Specifically, they are \( (A-6)+14, (A-6)+12, (A-6)+10, (A-6)+8 \) which gives us \( A+8, A+10, A+12, A+14 \).

3. **Calculating the New Average**:
Now, we have a total of 14 odd numbers:
- Previous three: \( A-8, A-6, A-4 \)
- Original seven: \( A-6, A-4, A-2, A, A+2, A+4, A+6 \)
- Next four: \( A+8, A+10, A+12, A+14 \)

The sum of these 14 numbers can be calculated as:
\[ \text{Sum} = (A-8) + (A-6) + (A-4) + (A-2) + A + (A+2) + (A+4) + (A+6) + (A+8) + (A+10) + (A+12) + (A+14) \]
Simplifying the sum:
\[ \text{Sum} = 14A + (-8 - 6 - 4 - 2 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14) \]
The terms in the parentheses sum to 0:
\[ \text{Sum} = 14A \]

4. **Finding the New Average**:
The new average \( A' \) of these 14 numbers is:
\[ A' = \frac{\text{Sum}}{14} = \frac{14A}{14} = A \]

### Final Answer:
The new average of these 14 consecutive odd numbers is \( A \).