The sum of five consecutive even numbers is equal to 260. What is the sum of the largest number amongst them and the half the square of the smallest number amongst them?

Options

A. 1644
B. 1208
C. 1346
D. 1288

Answer

To solve the problem, we need to find five consecutive even numbers whose sum is 260, and then calculate the sum of the largest number among them and half the square of the smallest number.

### Step-by-Step Solution:

1. **Define the Consecutive Even Numbers**:
Let the first even number be \( x \). The five consecutive even numbers can then be expressed as:
- First number: \( x \)
- Second number: \( x + 2 \)
- Third number: \( x + 4 \)
- Fourth number: \( x + 6 \)
- Fifth number: \( x + 8 \)

2. **Set Up the Equation**:
The sum of these five consecutive even numbers is given as 260:
\[ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 260 \]
Simplifying this, we get:
\[ 5x + 20 = 260 \]

3. **Solve for \( x \)**:
To isolate \( x \), subtract 20 from both sides:
\[ 5x = 260 - 20 \]
\[ 5x = 240 \]
Now, divide by 5:
\[ x = \frac{240}{5} = 48 \]

4. **Identify the Five Consecutive Even Numbers**:
Now that we have \( x = 48 \), the five consecutive even numbers are:
- 48 (first)
- 50 (second)
- 52 (third)
- 54 (fourth)
- 56 (fifth)

5. **Find the Largest and Smallest Numbers**:
The largest number among them is 56, and the smallest number is 48.

6. **Calculate Half the Square of the Smallest Number**:
First, calculate the square of the smallest number (48):
\[ 48^2 = 2304 \]
Now, find half of this value:
\[ \frac{2304}{2} = 1152 \]

7. **Sum the Largest Number and Half the Square of the Smallest Number**:
Now, we need to find the sum of the largest number (56) and half the square of the smallest number (1152):
\[ 56 + 1152 = 1208 \]

### Final Answer:
The sum of the largest number and half the square of the smallest number is **1208**.

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