The sum of the squares of three consecutive natural numbers is 77. Find the numbers.

Answer

To solve the problem of finding three consecutive natural numbers whose squares sum to 77, we can follow these steps:

### Step 1: Define the Variables
Let the three consecutive natural numbers be:
- First number: \( x - 1 \)
- Second 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 is equal to 77. Therefore, we can write the equation:
\[ (x - 1)^2 + x^2 + (x + 1)^2 = 77 \]

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

Now, substituting these expansions into the equation gives:
\[ (x^2 - 2x + 1) + x^2 + (x^2 + 2x + 1) = 77 \]

### Step 4: Combine Like Terms
Combining all the terms results in:
\[ x^2 - 2x + 1 + x^2 + x^2 + 2x + 1 = 77 \]
This simplifies to:
\[ 3x^2 + 2 = 77 \]

### Step 5: Isolate the Quadratic Term
Next, we will isolate the \( x^2 \) term:
\[ 3x^2 = 77 - 2 \]
\[ 3x^2 = 75 \]

### Step 6: Solve for \( x^2 \)
Now, divide both sides by 3:
\[ x^2 = \frac{75}{3} = 25 \]

### Step 7: Find \( x \)
Taking the square root of both sides gives:
\[ x = \sqrt{25} = 5 \]
Since we are looking for natural numbers, we only consider the positive root.

### Step 8: Determine the Consecutive Numbers
Now that we have \( x = 5 \), we can find the three consecutive natural numbers:
- First number: \( x - 1 = 5 - 1 = 4 \)
- Second number: \( x = 5 \)
- Third number: \( x + 1 = 5 + 1 = 6 \)

### Final Answer
The three consecutive natural numbers are **4, 5, and 6**.

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