The sum of the squares of three consecutive natural numbers is 110. Determine the numbers.

Answer

To solve the problem of finding three consecutive natural numbers whose squares sum to 110, we can follow these steps:

### Step 1: Define the Variables
Let the first consecutive natural number be \( x \). Therefore, the three consecutive natural numbers can be expressed as:
- First number: \( x \)
- Second number: \( x + 1 \)
- Third number: \( x + 2 \)

### Step 2: Set Up the Equation
We need to find the sum of the squares of these three numbers:
\[ x^2 + (x + 1)^2 + (x + 2)^2 = 110 \]

### Step 3: Expand the Squares
Now, we expand the squares in the equation:
\[ x^2 + (x^2 + 2x + 1) + (x^2 + 4x + 4) = 110 \]
Combining like terms gives:
\[ x^2 + x^2 + 2x + 1 + x^2 + 4x + 4 = 110 \]
This simplifies to:
\[ 3x^2 + 6x + 5 = 110 \]

### Step 4: Rearrange the Equation
Next, we rearrange the equation to set it to zero:
\[ 3x^2 + 6x + 5 - 110 = 0 \]
This simplifies to:
\[ 3x^2 + 6x - 105 = 0 \]

### Step 5: Simplify the Equation
We can simplify this equation by dividing all terms by 3:
\[ x^2 + 2x - 35 = 0 \]

### Step 6: Factor the Quadratic Equation
Now, we need to factor the quadratic equation:
\[ (x + 7)(x - 5) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 7 = 0 \) → \( x = -7 \) (not a natural number)
2. \( x - 5 = 0 \) → \( x = 5 \)

Since we are looking for natural numbers, we take \( x = 5 \).

### Step 8: Find the Three Consecutive Numbers
The three consecutive natural numbers are:
- First number: \( 5 \)
- Second number: \( 6 \)
- Third number: \( 7 \)

### Step 9: Verify the Solution
Now, we verify the solution by calculating the sum of their squares:
\[ 5^2 + 6^2 + 7^2 = 25 + 36 + 49 \]
Calculating this gives:
\[ 25 + 36 = 61 \]
\[ 61 + 49 = 110 \]
The sum is indeed 110, confirming our solution.

### Final Answer
The three consecutive natural numbers are **5, 6, and 7**.

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