The sum of the first \( n \) positive integers is 210. Find \( n \).
["## Finding ( n ): The Sum of the First ( n ) Positive Integers Equals 210", "Understanding arithmetic sequences is fundamental in mathematics, and one of the most common problems involves finding the sum of the first ( n ) positive integers. If you’ve ever calculated ( 1 + 2 + 3 + \ldots + n ) and discovered a neat formula, you’re already familiar with this concept.", "### The Formula for the Sum of the First ( n ) Positive Integers", "The sum of the first ( n ) positive integers is given by the well-known arithmetic series formula:", "[\nS_n = \frac{n(n + 1)}{2}\n]", "This formula calculates the total when adding all whole numbers from 1 to ( n ) in sequence. For example:", "- ( S_1 = \frac{1 \cdot 2}{2} = 1 )\n- ( S_2 = \frac{2 \cdot 3}{2} = 3 )\n- ( S_3 = \frac{3 \cdot 4}{2} = 6 ), and so on.", "### Solve for ( n ) When ( S_n = 210 )", "We are given that:", "[\n\frac{n(n + 1)}{2} = 210\n]", "Multiply both sides by 2 to eliminate the fraction:", "[\nn(n + 1) = 420\n]", "Now expand and rearrange into standard quadratic form:", "[\nn^2 + n - 420 = 0\n]", "This is a quadratic equation. Use the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "With ( a = 1 ), ( b = 1 ), and ( c = -420 ):", "[\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 ):", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n\quad \ ext{(discard negative root as ( n ) must be positive)}\n]", "### Conclusion", "The first positive integer ( n ) for which the sum of the first ( n ) positive integers equals 210 is:", "[\n\boxed{20}\n]", "This simple yet powerful formula not only solves the problem at hand but also lays the foundation for more advanced topics like series, averages, and algebra.", "---", "### Why This Matters", "Knowing how to find ( n ) in such equations comes in handy in problem-solving, programming, economics, and everyday financial planning—particularly when averaging or totaling sequential growth or counts.", "Understanding and applying the sum formula helps build logical thinking and mathematical fluency!"]
