The sum of the squares of three consecutive natural numbers is 2030. Then, what is the middle number?

Options

A. 25
B. 26
C. 27
D. 28

Answer

To solve the problem of finding the middle number of three consecutive natural numbers whose squares sum up to 2030, we can follow these steps:

### Step 1: Define the Consecutive Natural Numbers
Let the three consecutive natural numbers be represented as:
- First number: \( x - 1 \)
- Second number (middle number): \( x \)
- Third number: \( x + 1 \)

### Step 2: Set Up the Equation
According to the problem, the sum of the squares of these three numbers equals 2030. Therefore, we can write the equation:
\[ (x - 1)^2 + x^2 + (x + 1)^2 = 2030 \]

### Step 3: Expand the Squares
Now, we will expand each term in the equation:
\[ (x - 1)^2 = x^2 - 2x + 1 \]
\[ x^2 = x^2 \]
\[ (x + 1)^2 = x^2 + 2x + 1 \]

### Step 4: Combine the Expanded Terms
Now, substituting the expanded forms back into the equation:
\[ (x^2 - 2x + 1) + x^2 + (x^2 + 2x + 1) = 2030 \]
Combining like terms:
\[ x^2 - 2x + 1 + x^2 + x^2 + 2x + 1 = 2030 \]
This simplifies to:
\[ 3x^2 + 2 = 2030 \]

### Step 5: Isolate the Quadratic Term
Next, we will isolate the term with \( x^2 \):
\[ 3x^2 = 2030 - 2 \]
\[ 3x^2 = 2028 \]

### Step 6: Solve for \( x^2 \)
Now, divide both sides by 3:
\[ x^2 = \frac{2028}{3} \]
Calculating the division:
\[ x^2 = 676 \]

### Step 7: Find the Value of \( x \)
To find \( x \), we take the square root of both sides:
\[ x = \sqrt{676} \]
Calculating the square root gives:
\[ x = 26 \]

### Conclusion
Thus, the middle number is:
\[ \text{Middle number} = 26 \]