The sum of four consecutive even numbers is 44. What is the sum of the squares of these numbers?

Options

A. 288
B. 504
C. 696
D. 920

Answer

To solve the problem, we need to find four consecutive even numbers whose sum is 44, and then calculate the sum of their squares. Let's go through the steps:

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

### Step 2: Express the consecutive even numbers
The four consecutive even numbers can be expressed as:
- First number: \( x \)
- Second number: \( x + 2 \)
- Third number: \( x + 4 \)
- Fourth number: \( x + 6 \)

### Step 3: Set up the equation for their sum
According to the problem, the sum of these four numbers is 44. We can write this as:
\[ x + (x + 2) + (x + 4) + (x + 6) = 44 \]

### Step 4: Simplify the equation
Combining like terms, we have:
\[ 4x + 12 = 44 \]

### Step 5: Solve for \( x \)
To isolate \( x \), subtract 12 from both sides:
\[ 4x = 44 - 12 \]
\[ 4x = 32 \]
Now, divide both sides by 4:
\[ x = 8 \]

### Step 6: Find the four consecutive even numbers
Now that we have \( x \), we can find the four consecutive even numbers:
- First number: \( 8 \)
- Second number: \( 10 \) (which is \( 8 + 2 \))
- Third number: \( 12 \) (which is \( 8 + 4 \))
- Fourth number: \( 14 \) (which is \( 8 + 6 \))

### Step 7: Calculate the sum of the squares
Now, we need to calculate the sum of the squares of these numbers:
\[ 8^2 + 10^2 + 12^2 + 14^2 \]

Calculating each square:
- \( 8^2 = 64 \)
- \( 10^2 = 100 \)
- \( 12^2 = 144 \)
- \( 14^2 = 196 \)

Now, add these squares together:
\[ 64 + 100 + 144 + 196 = 504 \]

### Final Answer
The sum of the squares of these four consecutive even numbers is \( 504 \).

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