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, we need to find three consecutive natural numbers whose squares sum up to 2030. Let's denote the middle number as \( x \). Therefore, the three consecutive natural numbers can be expressed as:

1. First number: \( x - 1 \)
2. Middle number: \( x \)
3. Third number: \( x + 1 \)

Now, we can set up the equation based on the information given in the question:

\[ (x - 1)^2 + x^2 + (x + 1)^2 = 2030 \]

Now, let's expand and simplify the left-hand side of the equation:

1. Expanding \( (x - 1)^2 \):
\[ (x - 1)^2 = x^2 - 2x + 1 \]

2. Expanding \( (x + 1)^2 \):
\[ (x + 1)^2 = x^2 + 2x + 1 \]

Now, substituting these expansions 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:

\[ 3x^2 + 2 = 2030 \]

Next, we will isolate \( x^2 \):

\[ 3x^2 = 2030 - 2 \]
\[ 3x^2 = 2028 \]

Now, divide both sides by 3:

\[ x^2 = \frac{2028}{3} \]
\[ x^2 = 676 \]

Taking the square root of both sides gives us:

\[ x = \sqrt{676} \]
\[ x = 26 \]

Thus, the middle number is \( 26 \).

### Summary of Steps:
1. Define the three consecutive numbers in terms of \( x \).
2. Set up the equation for the sum of their squares.
3. Expand the squares and combine like terms.
4. Solve for \( x^2 \) and then find \( x \).