The sum of the first \( n \) terms of an arithmetic sequence is given by \( S_n = \frac{n}{2}(2a + (n-1)d) \). If the first term is 3 and the common difference is 2, find \( n \) such that \( S_n = 210 \).
["Understanding the Sum of an Arithmetic Sequence: Solving for ( n ) When ( S_n = 210 )", "When studying arithmetic sequences, one of the most essential formulas is that of the sum of the first ( n ) terms:", "[\nS_n = \frac{n}{2} \left(2a + (n - 1)d \right)\n]", "This formula allows us to quickly calculate the total sum based on the first term ( a ), the common difference ( d ), and the number of terms ( n ).", "---", "### Applying the Formula to Real Problems", "Let’s apply this formula to a common problem Students often encounter:\nIf the first term ( a = 3 ), the common difference ( d = 2 ), and the sum ( S_n = 210 ), find the value of ( n ).", "### Step 1: Substitute known values into the formula", "Given:\n- ( a = 3 )\n- ( d = 2 )\n- ( S_n = 210 )", "Substitute into the sum formula:", "[\n210 = \frac{n}{2} \left(2(3) + (n - 1)(2) \right)\n]", "### Step 2: Simplify the expression inside the parentheses", "Calculate step by step:", "[\n210 = \frac{n}{2} \left(6 + 2(n - 1)\right)\n]\n[\n210 = \frac{n}{2} \left(6 + 2n - 2\right)\n]\n[\n210 = \frac{n}{2} \left(2n + 4\right)\n]", "### Step 3: Simplify the equation", "Factor inside the parentheses:", "[\n210 = \frac{n}{2} \cdot 2(n + 2)\n]", "Simplify:", "[\n210 = n(n + 2)\n]", "### Step 4: Solve the quadratic equation", "Expand the right side:", "[\nn^2 + 2n - 210 = 0\n]", "Factor or use the quadratic formula. Since this factors neatly:", "Find two numbers that multiply to ( -210 ) and add to ( 2 ): ( 15 ) and ( -14 )", "[\n(n + 15)(n - 14) = 0\n]", "Solutions: ( n = -15 ) or ( n = 14 )", "Since ( n ) represents the number of terms, it must be a positive integer:", "[\nn = 14\n]", "---", "### Conclusion", "By applying the arithmetic series sum formula and solving a simple quadratic equation, we find that when the first term is 3 and the common difference is 2, the number of terms ( n ) required for the sum to equal 210 is:", "[\n\boxed{14}\n]", "Understanding this method empowers learners to tackle related problems efficiently and strengthens their foundation in algebra and sequences.", "---", "Keywords: arithmetic sequence sum formula, ( S_n = \frac{n}{2}(2a + (n-1)d) ), arithmetic series problem, solve for ( n ), first term 3, common difference 2, quadratic equation, sum of arithmetic terms."]
