If the sum of the squares of three consecutive natural numbers is 110, then the smallest of these natural numbers is :

Options

A. 8
B. 6
C. 7
D. 5

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Define the Variables
Let the three consecutive natural numbers be:
- The first number: \( x - 1 \)
- The second number: \( x \)
- The third number: \( x + 1 \)

### Step 2: Write the Equation
According to the problem, the sum of the squares of these three numbers is equal to 110. Therefore, we can write the equation as:
\[ (x - 1)^2 + x^2 + (x + 1)^2 = 110 \]

### 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, substitute the expanded forms back into the equation:
\[ (x^2 - 2x + 1) + x^2 + (x^2 + 2x + 1) = 110 \]
Combining like terms:
\[ x^2 - 2x + 1 + x^2 + x^2 + 2x + 1 = 110 \]
This simplifies to:
\[ 3x^2 + 2 = 110 \]

### Step 5: Solve for \( x^2 \)
Now, we will isolate \( x^2 \):
\[ 3x^2 = 110 - 2 \]
\[ 3x^2 = 108 \]
Now, divide both sides by 3:
\[ x^2 = \frac{108}{3} = 36 \]

### Step 6: Find \( x \)
Now, take the square root of both sides:
\[ x = \sqrt{36} = 6 \]

### Step 7: Find the Smallest Natural Number
Since \( x \) is the middle number, the smallest natural number is:
\[ x - 1 = 6 - 1 = 5 \]

### Final Answer
The smallest of the three consecutive natural numbers is:
\[ \boxed{5} \]

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