\[ (n-1)^2 + n^2 + (n+1)^2 = 425 \]

["Title: Solving the Equation ((n-1)^2 + n^2 + (n+1)^2 = 425) – Step-by-Step Guide", "Mathematics often reveals elegant solutions through seemingly simple equations. One such intriguing problem is solving:\n((n-1)^2 + n^2 + (n+1)^2 = 425)", "In this SEO-optimized article, we’ll explore how to simplify this expression, derive a formula for (n), and understand its significance—perfect for students, math enthusiasts, and problem solvers.", "---", "### How to Simplify ((n-1)^2 + n^2 + (n+1)^2 = 425) – Step-by-Step Breakdown", "The equation combines three squared terms spaced around (n). Let’s expand each term:", "1. Expand ((n-1)^2 = n^2 - 2n + 1)\n2. Expand (n^2) remains just (n^2)\n3. Expand ((n+1)^2 = n^2 + 2n + 1)", "Now, add all terms together:", "[\n(n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1) = 425\n]", "Combine like terms:", "- (n^2 + n^2 + n^2 = 3n^2)\n- (-2n + 2n = 0) (n terms cancel out)\n- (1 + 1 = 2)", "So the equation becomes:\n[\n3n^2 + 2 = 425\n]", "---", "### Solve for (n) Mathematically", "Subtract 2 from both sides:\n[\n3n^2 = 423\n]", "Divide by 3:\n[\nn^2 = 141\n]", "Take the square root:\n[\nn = \sqrt{141} \quad \ ext{or} \quad n = -\sqrt{141}\n]", "Since (\sqrt{141} \approx 11.87) is not an integer, this equation has no integer solutions—yet it offers valuable insights.", "---", "### Understanding the Structure: Why This Equation Matters", "Even though (n) is irrational, the structure ((n-1)^2 + n^2 + (n+1)^2) appears frequently in geometry and number theory—especially in problems involving distances on a number line, lattice points, or optimization.", "For example:", "- It represents the sum of squared distances from point (n) to points (-1), (0), and (1) on the real line.\n- In coordinate geometry, similar expressions model squared distances in 1D space.\n- Compounding such expressions leads to rules and formulas useful in algebra and data analysis.", "---", "### Derive a General Formula from ((n-1)^2 + n^2 + (n+1)^2 = C)", "From our simplification:", "[\n3n^2 + 2 = C \quad \Rightarrow \quad n^2 = \frac{C - 2}{3}\n]", "Thus, for any constant (C), the solution for (n) is:\n[\nn = \pm \sqrt{\frac{C - 2}{3}}\n]", "This formula is helpful when solving variation problems where three symmetric values around (n) generate a total.", "---", "### Practical Applications and Real-World Relevance", "- Physics & Engineering: Sum-of-squares expressions model energy, error margins, and signal variance.\n- Data Science: Used in least-squares regression and error analysis.\n- Geometry: Helps compute distances in discrete grid systems.", "Even when (n) isn’t an integer, the formula reveals symmetries and relationships critical for deeper mathematical understanding.", "---", "### Final Thoughts", "While ((n-1)^2 + n^2 + (n+1)^2 = 425) does not yield a simple integer (n), the equation beautifully illustrates how symmetry and algebra combine to solve complex layouts. Mastering such problems strengthens problem-solving skills and prepares learners for advanced studies in algebra, calculus, and applied sciences.", "If you’re exploring patterns with three-term symmetric sums, this equation serves as a foundational gem—simple to solve, profound in application.", "---", "### SEO Keywords Included\n- Solve ((n-1)^2 + n^2 + (n+1)^2 = 425`\n- Algebraic equation solving steps\n- Mathematical patterns in number theory\n- Optimization with symmetric expressions\n- Useful formulas in 1D geometry\n- Finding integer and irrational solutions", "---", "Want more square-sum puzzles? Explore how (a^2 + b^2 + c^2) expands in sequences, or dive deeper into quadratic identities and symmetry-based problem solving.", "---", "*Keywords: (n-1)^2 + n^2 + (n+1)^2 = 425, algebra, equation solving, math tutorial, quadratic reasoning, number patterns, symmetry in math, sum of squares formula, irrational solutions, educational math."]