Sum: (n−1)² + n² + (n+1)² = n²−2n+1 + n² + n²+2n+1 = 3n² + 2
["Mastering the Sum: (n−1)² + n² + (n+1)² Simplified", "Mathematics thrives on patterns, and one elegant example is the simplification of the sum:\n(n−1)² + n² + (n+1)²", "At first glance, expanding each squared term looks tedious—but there’s a smarter way to understand it, revealing the clean result:\n3n² + 2", "---", "### Unpacking the Expression", "Let’s expand each term individually:", "- ( (n−1)² = n² − 2n + 1 )\n- ( n² = n² )\n- ( (n+1)² = n² + 2n + 1 )", "Now, add them together:", "[\n(n−1)² + n² + (n+1)² = (n² − 2n + 1) + n² + (n² + 2n + 1)\n]", "---", "### Combine Like Terms", "Group the like components:", "- ( n² + n² + n² = 3n² )\n- ( −2n + 2n = 0 ) (the linear terms cancel out)\n- ( 1 + 1 = 2 )", "This gives:", "[\n3n² + 2\n]", "---", "### Why This Identity Matters", "This identity showcases a symmetrical algebraic pattern—the sum of three consecutive squares centered on ( n ). Such patterns simplify real-world problem solving, from probability to computer science, where knowing Σ(n±1)² in terms of n accelerates calculations.", "It also emphasizes the beauty of algebraic symmetry and confirmation via expansion.", "---", "Conclusion", "The sum ( (n−1)² + n² + (n+1)² ) simplifies beautifully to 3n² + 2, demonstrating how pattern recognition and algebraic expansion work hand in hand. Whether for homework, exams, or coding, mastering such identities enhances both speed and insight.", "---", "Keywords: (n−1)² + n² + (n+1)², simplification, algebraic sum, pattern recognition, n squared identity, 3n² + 2, math simplification, algebra tutorial, quadratic expressions", "Meta Description:\nDiscover how (n−1)² + n² + (n+1)² simplifies to 3n² + 2 using algebraic expansion and symmetry—ideal for mastering quadratic identities in mathematics."]
