There are 6 consecutive odd numbers. The square of the average of the last three numbers is 386 more than the product of the first two numbers. What is the value of the first odd number?

Options

A. 25
B. 23
C. 27
D. 29

Answer

To solve the problem step by step, we will follow the logical reasoning presented in the video transcript:

### Step 1: Define the first odd number
Let the first odd number be \( x \).

### Step 2: List the six consecutive odd numbers
The six consecutive odd numbers can be expressed as:
- First: \( x \)
- Second: \( x + 2 \)
- Third: \( x + 4 \)
- Fourth: \( x + 6 \)
- Fifth: \( x + 8 \)
- Sixth: \( x + 10 \)

### Step 3: Calculate the average of the last three numbers
The last three numbers are:
- Fourth: \( x + 6 \)
- Fifth: \( x + 8 \)
- Sixth: \( x + 10 \)

The sum of these three numbers is:
\[ (x + 6) + (x + 8) + (x + 10) = 3x + 24 \]
The average of these three numbers is:
\[ \text{Average} = \frac{3x + 24}{3} = x + 8 \]

### Step 4: Square the average
Now, we will square the average:
\[ \text{Square of the average} = (x + 8)^2 = x^2 + 16x + 64 \]

### Step 5: Calculate the product of the first two numbers
The product of the first two numbers is:
\[ x \cdot (x + 2) = x^2 + 2x \]

### Step 6: Set up the equation
According to the problem, the square of the average of the last three numbers is 386 more than the product of the first two numbers. Thus, we can set up the equation:
\[ x^2 + 16x + 64 = x^2 + 2x + 386 \]

### Step 7: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ 16x + 64 = 2x + 386 \]
Now, subtract \( 2x \) from both sides:
\[ 14x + 64 = 386 \]
Next, subtract 64 from both sides:
\[ 14x = 386 - 64 \]
\[ 14x = 322 \]

### Step 8: Solve for \( x \)
Now, divide both sides by 14:
\[ x = \frac{322}{14} = 23 \]

### Conclusion
The value of the first odd number is \( 23 \).