Relationships Between Squares
Posted on: April 26, 2025
Interesting relationships between squares based on the relative size of the numbers:
Case 1: When x > n
\[ n^2 = x^2 + (x-n) - 2\left[(n+1) + (n+2) + \ldots + x\right] \]
Case 2: When x < n
\[ n^2 = x^2 + (x-n) + 2\left[n + (n-1) + \ldots + (x+1)\right] \]
Let's verify these formulas with some examples:
Example 1: Let n = 3 and x = 5 (Case 1: x > n)
\begin{align}
3^2 &= 5^2 + (5-3) - 2\left[(3+1) + (3+2) + 5\right] \\
9 &= 25 + 2 - 2\left[4 + 5 + 5\right] \\
9 &= 27 - 2 \cdot 14 \\
9 &= 27 - 28 \\
9 &= 9
\end{align}
Example 2: Let n = 5 and x = 2 (Case 2: x < n)
\begin{align}
5^2 &= 2^2 + (2-5) + 2\left[5 + 4 + 3\right] \\
25 &= 4 - 3 + 2\left[12\right] \\
25 &= 1 + 24 \\
25 &= 25
\end{align}