Set \( \frac{n(n+1)}{2} = 210 \). Multiply through by 2: \( n(n+1) = 420 \).

["# Solving the Equation ( \frac{n(n+1)}{2} = 210 ): A Step-by-Step Guide", "The equation", "[\n\frac{n(n+1)}{2} = 210\n]", "represents a classic problem in number theory and algebra: finding a positive integer ( n ) such that the sum of the first ( n ) natural numbers equals 210. This type of equation arises frequently in combinatorics and algorithmic problems, particularly when analyzing triangular numbers.", "## Why Triangular Numbers Matter", "The expression ( \frac{n(n+1)}{2} ) defines the ( n^{\ ext{th}} ) triangular number — the total count of objects arranged in a triangular formation. Understanding how to solve such equations helps in grasping patterns in sequences, optimizing algorithms, and solving real-world counting problems.", "## Step 1: Eliminate the Fraction by Multiplying Through", "To simplify the equation, eliminate the denominator by multiplying both sides by 2:", "[\nn(n+1) = 420\n]", "This step transforms the equation into a quadratic form, making it easier to solve using standard algebraic methods.", "## Step 2: Expand and Rearrange into a Standard Quadratic Form", "Expanding the left-hand side gives:", "[\nn^2 + n = 420\n]", "Rearranging into standard quadratic form:", "[\nn^2 + n - 420 = 0\n]", "This is a quadratic equation in the form ( an^2 + bn + c = 0 ), with ( a = 1 ), ( b = 1 ), and ( c = -420 ).", "## Step 3: Solve the Quadratic Equation", "We solve this quadratic equation using the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substituting the values:", "[\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-420)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 1680}}{2} = \frac{-1 \pm \sqrt{1681}}{2}\n]", "Since ( \sqrt{1681} = 41 ), we have:", "[\nn = \frac{-1 \pm 41}{2}\n]", "This gives two potential solutions:", "[\nn = \frac{-1 + 41}{2} = 20 \quad \ ext{and} \quad n = \frac{-1 - 41}{2} = -21\n]", "## Step 4: Interpret the Solution", "Because ( n ) represents a natural number (count of items), we discard the negative solution. Thus, the only valid solution is:", "[\nn = 20\n]", "We verify:", "[\n\frac{20(20+1)}{2} = \frac{20 \ imes 21}{2} = \frac{420}{2} = 210\n]", "The result checks.", "## Conclusion", "The solution to the equation ( \frac{n(n+1)}{2} = 210 ) is ( n = 20 ). This demonstrates how a simple formula for triangular numbers leads cleanly to a quadratic equation, solvable via well-established algebraic techniques. Mastering such problems strengthens logical reasoning and is essential for both academic and practical mathematical applications.", "---", "Key Takeaways:", "- Multiply both sides by 2 to eliminate the denominator.\n- Rewrite as a quadratic equation for standard solving methods.\n- Use the quadratic formula and discard invalid (negative) solutions.\n- Verify the solution to ensure correctness.", "Whether you’re a student learning algebra or a developer applying mathematical reasoning, equations like ( \frac{n(n+1)}{2} = 210 ) offer valuable insight into number patterns and solution strategies."]