If the sum of squares of three consecutive natural number is 2702. Then, what will be the middle number ?

Options

A. 29
B. 30
C. 31
D. 32

Answer

To solve the problem of finding the middle number among three consecutive natural numbers whose sum of squares equals 2702, we can follow these steps:

### Step 1: Define the Variables
Let's denote the three consecutive natural numbers as:
- The first number: \( x - 1 \)
- The second number (middle number): \( x \)
- The third number: \( x + 1 \)

### Step 2: Write the Equation for the Sum of Squares
The sum of the squares of these three numbers can be expressed as:
\[ (x - 1)^2 + x^2 + (x + 1)^2 = 2702 \]

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

### Step 4: Combine the Expanded Terms
Adding these expanded terms together gives:
\[ (x^2 - 2x + 1) + x^2 + (x^2 + 2x + 1) = 2702 \]
This simplifies to:
\[ 3x^2 + 2 = 2702 \]

### Step 5: Solve for \( x^2 \)
Now, we will isolate \( x^2 \):
\[ 3x^2 = 2702 - 2 \]
\[ 3x^2 = 2700 \]
\[ x^2 = \frac{2700}{3} = 900 \]

### Step 6: Find \( x \)
Taking the square root of both sides, we get:
\[ x = \sqrt{900} = 30 \]

### Step 7: Identify the Middle Number
The middle number, which we defined as \( x \), is:
\[ \text{Middle Number} = 30 \]

### Final Answer
The middle number is **30**.

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