However, double-check: maybe \( a=3, d=2 \), \( S_n = \frac{n}{2}[6 + (n-1)2] = \frac{n}{2}(2n + 4) = n(n+2) \).

["Title: Step-by-Step Verification of the Formula for the Sum ( S_n )", "---", "SEO Meta Description:\nCarefully verify the formula ( S_n = \frac{n}{2}[6 + (n-1) \cdot 2] ) and confirm it simplifies to ( S_n = n(n + 2) ). This essential arithmetic derivation is crucial for understanding summation patterns.", "---", "When studying arithmetic sequences, one common challenge is verifying or simplifying sum formulas algebraically. Suppose you encounter the expression for the sum ( S_n ) defined as:\n[\nS_n = \frac{n}{2} \left[6 + (n - 1) \cdot 2\right]\n]\nAt first glance, this may appear complex, but a careful double-check confirms a straightforward simplification. Let’s walk through the derivation step-by-step to verify the claim:\n[\nS_n = n(n + 2)\n]", "---", "Step 1: Expand the expression inside the brackets\nBegin by simplifying the term inside the parentheses:\n[\n6 + (n - 1) \cdot 2\n]\nMultiply 2 by ( n - 1 ):\n[\n6 + 2n - 2\n]\nCombine like terms:\n[\n2n + 4\n]", "---", "Step 2: Substitute back into the full formula\nReplace the bracketed expression with ( 2n + 4 ):\n[\nS_n = \frac{n}{2} (2n + 4)\n]", "---", "Step 3: Factor and simplify\nFactor 2 inside the parentheses:\n[\nS_n = \frac{n}{2} \cdot 2(n + 2)\n]\nCancel the ( 2 ) in the numerator and denominator:\n[\nS_n = n(n + 2)\n]", "---", "Conclusion:\nThrough careful algebraic expansion and simplification, we have verified that\n[\nS_n = \frac{n}{2} \left[6 + (n - 1) \cdot 2\right] \quad \ ext{is indeed equal to} \quad S_n = n(n + 2)\n]\nThis confirms the validity of the derived formula, making it a powerful tool for computing sums in arithmetic progressions.", "---", "Further Reading:\nLearn how summation notation (n-sum) underpins sequence analysis and why recognizing such formulas accelerates problem-solving in algebra and discrete mathematics.", "---", "Keywords:\narithmetic series sum, verify ( S_n ), confirm formula ( S_n = n(n+2) ), algebra simplification, summation formula, arithmetic sequence, ( a = 3 ), ( d = 2 ) verification", "---", "Note:\nAlways double-check formulas by expanding and simplifying — it’s a reliable method to uncover hidden patterns. Proper verification strengthens your confidence in mathematical derivations."]