\[ S_n = rac{n}{2}(2 imes 3 + (n-1) imes 2) = 210 \]

["# Solving the Equation: ( S_n = \frac{n}{2}(2 \ imes 3 + (n-1) \ imes 2) = 210 )", "If you’ve ever encountered a formula like ( S_n = \frac{n}{2}(2 \ imes 3 + (n-1) \ imes 2) = 210 ), you’re likely tackling a classic problem rooted in arithmetic sequences. In this article, we’ll break down this expression, explain how to solve it, and uncover the value of ( n )—all while optimizing for search engines to help you understand and work through similar mathematical challenges.", "---", "## Understanding the Formula: Sum of an Arithmetic Sequence", "The given expression:", "[\nS_n = \frac{n}{2}(2a + (n-1)d) = 210\n]", "represents the sum of the first ( n ) terms of an arithmetic sequence, where:\n- ( S_n ) is the sum of the first ( n ) terms\n- ( a ) is the first term\n- ( d ) is the common difference\n- ( n ) is the number of terms", "In your equation, the first term ( a = 2 \ imes 3 = 6 ), and the common difference ( d = 2 ). There are 3 initial terms, so the formula becomes:\n[\nS_n = \frac{n}{2} \left( 2 \cdot 6 + (n-1) \cdot 2 \right) = 210\n]", "---", "## Simplify the Expression", "First, compute the constants inside the parentheses:", "[\n2 \cdot 6 = 12\n]\n[\n(n - 1) \cdot 2 = 2n - 2\n]", "Now add them:", "[\n12 + 2n - 2 = 2n + 10\n]", "Substitute back into the sum formula:", "[\nS_n = \frac{n}{2}(2n + 10) = 210\n]", "Factor numerator:", "[\nS_n = \frac{n \cdot 2(n + 5)}{2} = 210\n]", "Simplify:", "[\nn(n + 5) = 210\n]", "---", "## Solve the Quadratic Equation", "Now you have:", "[\nn^2 + 5n = 210\n]\n[\nn^2 + 5n - 210 = 0\n]", "Apply the quadratic formula:", "[\nn = \frac{ -5 \pm \sqrt{5^2 - 4(1)(-210)} }{2}\n]\n[\nn = \frac{ -5 \pm \sqrt{25 + 840} }{2}\n]\n[\nn = \frac{ -5 \pm \sqrt{865} }{2}\n]", "Wait—approximately, ( \sqrt{865} \approx 29.41 ), so:", "[\nn \approx \frac{ -5 + 29.41 }{2} = \frac{24.41}{2} \approx 12.205\n]", "This is not an integer, which contradicts the expectation that ( n ) should be a natural number (as it counts terms).", "---", "## Re-examining the Problem Assumptions", "Since ( \sqrt{865} ) is not a perfect square, we must reconsider: Did we interpret the sequence correctly?", "Let’s clarify:\nThe problem says:\n[\nS_n = \frac{n}{2}(2 \ imes 3 + (n-1) \ imes 2) = 210\n]\nwith first term ( a = 6 ), ( d = 2 ), which is correct.", "But the derived equation yields non-integer ( n ), suggesting either:", "- A typo in the sum value (210),\n- A misinterpretation of the sequence’s structure, or\n- The equation is correctly formed but has no integer solution under these constraints.", "---", "## Possible Corrections and Alternative Solutions", "### Case 1: Typo in the Result\nSuppose the sum was intended to be ( S_n = 210 ), but the sum yields a non-integer solution—then either:\n- Use ( S_n = 220 ) or ( S_n = 204 ), both of which give nice integer roots.", "For example, try ( n(n+5) = 220 ):", "[\nn^2 + 5n - 220 = 0\n]", "[\nn = \frac{ -5 \pm \sqrt{25 + 880} }{2} = \frac{ -5 \pm \sqrt{905} }{2} \approx 11.2\n]", "Still not perfect. Try ( S_n = 210 ) with adjusted ((n-1)):", "Suppose the average term formula was misapplied—sometimes the index starts differently.", "### Case 2: Recheck the Sequence Definition", "If instead the sequence is not starting at 6 with ( d = 2 ), but instead generalizes:\nLet the sum be ( S_n = \frac{n}{2}[2a + (n-1)d] = 210 ) with ( a = 3? ), ( d = 2? ), but this deviates.", "---", "## Using Algebra to Find Integer Solutions (General Approach)", "For ( S_n = \frac{n}{2}(2a + (n-1)d) = 210 ), with ( a = 6 ), ( d = 2 ), the equation is fixed.", "But suppose we treat ( a = 3 ), ( d = 2 ), or other values? The problem states clearly ( 2 \ imes 3 = 6 ) as first term, so ( a = 6 ) is correct.", "Try solving ( n(n + 5) = 210 )", "List squares near 210:", "15² = 225\n14² = 196", "Try factoring:", "( n^2 + 5n - 210 = 0 )", "Discriminant: ( 25 + 840 = 865 ), not a perfect square → no integer ( n )", "---", "## Conclusion: No Natural Solution Under Given Conditions", "The equation\n[\nS_n = \frac{n}{2}(12 + 2(n - 1)) = 210\n]\nsimplifies to\n[\nn(n + 5) = 210\n]\nwith no natural number solution for ( n ). Therefore, either:\n- The intended sum was incorrect,\n- The sequence parameters need reinterpretation, or\n- Additional context is required (e.g., partial sums, modified indexing).", "---", "## Practical Takeaways for Problem Solvers", "- Double-check initial values: Ensure ( a ), ( d ), and number of terms are correctly applied.\n- Validate discriminant: A positive, perfect-square discriminant is essential for integer solutions in quadratic sums.\n- Test nearby sums: If stuck, try ( S_n = 204, 210, 220 ) to find integer ( n ).\n- Use online quadratic solvers: Tools like symbols.lanxess.com help verify roots.", "---", "## Further Reading", "- Arithmetic Sequences — Wolfram MathWorld\n- Solving Quadratic Equations Step-by-Step\n- Sum of an Arithmetic Series—Stanford University", "---", "If you’re working on a similar equation, verify input values and simplify step-by-step. Whether ( n ) is integer depends entirely on consistent definitions. Stay precise—mathematics rewards careful expression and logical validation.", "---", "Keywords: arithmetic series sum, solve for ( n ), quadratic equation in ( n ), sum formula derivation, no integer solution, ( S_n = 210 ), arithmetic sequence problem"]