The sum of the squares of three consecutive natural numbers is $2030$. What is the middle number?

### Concept & Logic When solving for consecutive numbers involving squares, defining the sequence symmetrically around a middle term $x$ (e.g., $x-1, x, x+1$) eliminates the middle linear terms during expansion, making the quadratic equation vastly simpler to solve. ### Step-by-Step Solution * **Given:** Three consecutive natural numbers. Let them be $(x-1)$, $x$, and $(x+1)$. Here, $x$ represents the middle number. * **Condition:** The sum of their squares is $2030$. $$(x-1)^2 + x^2 + (x+1)^2 = 2030$$ * Expand the squared binomials: $$(x^2 - 2x + 1) + x^2 + (x^2 + 2x + 1) = 2030$$ * Combine like terms. Notice how the $-2x$ and $+2x$ neatly cancel out: $$3x^2 + 2 = 2030$$ * Subtract $2$ from both sides: $$3x^2 = 2028$$ * Divide by $3$: $$x^2 = 676$$ * Take the square root to find $x$: $$x = \sqrt{676} = 26$$ * The middle number $x$ is exactly $26$. ### Exam Strategy & Shortcut Use **Approximation**. The sum of the three squares is $2030$. Since the numbers are consecutive, they are very close to each other in value. Therefore, $3 \times (\text{Middle Number})^2 \approx 2030$. $(\text{Middle Number})^2 \approx \frac{2030}{3} \approx 676.6$. Looking at the options, we need the square root of a number very close to $676$. If you memorize squares up to $30$, you will instantly know that $26^2 = 676$. Option (b) is the clear winner without doing any algebraic expansion. ### Common Pitfall Setting the numbers up as $x$, $x+1$, and $x+2$ creates a much harder quadratic: $x^2 + (x^2 + 2x + 1) + (x^2 + 4x + 4) = 2030$, leading to $3x^2 + 6x - 2025 = 0 \Rightarrow x^2 + 2x - 675 = 0$. Factoring large quadratics under pressure often leads to errors. Always use symmetrical variables like $(x-1, x, x+1)$. ### Final Answer **Therefore, the correct answer is 26.**