The average of 7 consecutive odd numbers if the smallest of those numbers is denoted by k:

Options

A. k+4
B. k+7
C. k+6
D. 7k

Answer

To find the average of 7 consecutive odd numbers when the smallest number is denoted by \( k \), we can follow these steps:

### Step 1: Identify the 7 consecutive odd numbers
The smallest odd number is \( k \). The next consecutive odd numbers can be expressed as:
- First number: \( k \)
- Second number: \( k + 2 \)
- Third number: \( k + 4 \)
- Fourth number: \( k + 6 \)
- Fifth number: \( k + 8 \)
- Sixth number: \( k + 10 \)
- Seventh number: \( k + 12 \)

### Step 2: Write out the sum of these numbers
Now, we can sum these 7 numbers:
\[ \text{Sum} = k + (k + 2) + (k + 4) + (k + 6) + (k + 8) + (k + 10) + (k + 12) \]

### Step 3: Simplify the sum
Combine like terms:
\[ \text{Sum} = k + k + k + k + k + k + k + (2 + 4 + 6 + 8 + 10 + 12) \]
This simplifies to:
\[ \text{Sum} = 7k + (2 + 4 + 6 + 8 + 10 + 12) \]
Calculating the sum of the constants:
\[ 2 + 4 + 6 + 8 + 10 + 12 = 42 \]
Thus, the total sum is:
\[ \text{Sum} = 7k + 42 \]

### Step 4: Calculate the average
To find the average, we divide the total sum by the number of terms (which is 7):
\[ \text{Average} = \frac{7k + 42}{7} \]

### Step 5: Simplify the average
Now, we can simplify the average:
\[ \text{Average} = \frac{7k}{7} + \frac{42}{7} = k + 6 \]

### Final Answer
The average of the 7 consecutive odd numbers, with the smallest number denoted by \( k \), is:
\[ \boxed{k + 6} \]