The sum of four consecutive even number is 60. What is the sum of the squares of these numbers ?

Options

A. 1545
B. 920
C. 2015
D. 2136

Answer

To solve the problem of finding the sum of the squares of four consecutive even numbers that add up to 60, we can follow these steps:

### Step-by-Step Solution:

1. **Define the First Even Number**:
Let the first even number be represented as \( x \).

2. **Identify 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: \( x + 6 \)

3. **Set Up the Equation**:
According to the problem, the sum of these four numbers equals 60. Therefore, we can write the equation:
\[ x + (x + 2) + (x + 4) + (x + 6) = 60 \]

4. **Simplify the Equation**:
Combine like terms:
\[ 4x + 12 = 60 \]

5. **Solve for \( x \)**:
Subtract 12 from both sides:
\[ 4x = 60 - 12 \]
\[ 4x = 48 \]
Now, divide by 4:
\[ x = 12 \]

6. **List the Four Even Numbers**:
Now that we have \( x \), we can find the four consecutive even numbers:
- First number: \( 12 \)
- Second number: \( 14 \)
- Third number: \( 16 \)
- Fourth number: \( 18 \)

7. **Calculate the Squares of Each Number**:
Now we will find the squares of each of these numbers:
- \( 12^2 = 144 \)
- \( 14^2 = 196 \)
- \( 16^2 = 256 \)
- \( 18^2 = 324 \)

8. **Sum the Squares**:
Finally, we add these squares together:
\[ 144 + 196 + 256 + 324 \]

Performing the addition step by step:
- \( 144 + 196 = 340 \)
- \( 340 + 256 = 596 \)
- \( 596 + 324 = 920 \)

9. **Final Answer**:
The sum of the squares of the four consecutive even numbers is \( 920 \).