The average of three consecutive even numbers is A. If next five even numbers are added, what is the average of these eight numbers?

Options

A. `A+3`
B. `A+4`
C. `A+5`
D. `A+7`

Answer

To solve the problem step by step, we will first define the three consecutive even numbers and then find the average of the eight even numbers after adding the next five even numbers.

### Step 1: Define the three consecutive even numbers
Let the first even number be \( X \). Therefore, the three consecutive even numbers can be expressed as:
- First number: \( X \)
- Second number: \( X + 2 \)
- Third number: \( X + 4 \)

### Step 2: Calculate the average of the three consecutive even numbers
The average \( A \) of these three numbers is given by:
\[ A = \frac{X + (X + 2) + (X + 4)}{3} \]
Simplifying this, we get:
\[ A = \frac{3X + 6}{3} = X + 2 \]

### Step 3: Solve for \( X \)
From the equation \( A = X + 2 \), we can express \( X \) in terms of \( A \):
\[ X = A - 2 \]

### Step 4: Identify the next five consecutive even numbers
The next five even numbers after \( X + 4 \) are:
- Fourth number: \( X + 6 \)
- Fifth number: \( X + 8 \)
- Sixth number: \( X + 10 \)
- Seventh number: \( X + 12 \)
- Eighth number: \( X + 14 \)

### Step 5: Calculate the sum of all eight numbers
Now, we will sum all eight even numbers:
\[ \text{Sum} = X + (X + 2) + (X + 4) + (X + 6) + (X + 8) + (X + 10) + (X + 12) + (X + 14) \]
Combining like terms:
\[ \text{Sum} = 8X + (2 + 4 + 6 + 8 + 10 + 12 + 14) = 8X + 56 \]

### Step 6: Calculate the average of the eight numbers
The average of these eight numbers is:
\[ \text{Average} = \frac{\text{Sum}}{8} = \frac{8X + 56}{8} = X + 7 \]

### Step 7: Substitute \( X \) back in terms of \( A \)
Now, substitute \( X \) with \( A - 2 \):
\[ \text{Average} = (A - 2) + 7 = A + 5 \]

### Final Answer
Thus, the average of the eight consecutive even numbers is:
\[ \text{Average} = A + 5 \]