The sum of the first \(n\) positive integers is 210. What is \(n\)?
["The Sum of the First (n) Positive Integers Is 210: How to Find (n)", "Have you ever wondered how to calculate the sum of the first (n) positive integers when given a specific total—like when that sum equals 210? Knowing that the sum of the first (n) positive integers is 210 opens up a beautifully simple algebraic solution rooted in arithmetic series formulas. In this article, we’ll explore the sum formula, solve for (n), and understand the elegant math behind this classic problem.", "---", "### Understanding the Sum of the First (n) Positive Integers", "The sum of the first (n) positive integers is given by the well-known formula:", "[\nS = \frac{n(n + 1)}{2}\n]", "This formula appears in countless math problems and appears naturally when summing sequences like 1 + 2 + 3 + … + (n). It stems from the arithmetic series sum formula, where each term increases uniformly.", "Setting this sum equal to 210, we form the equation:", "[\n\frac{n(n + 1)}{2} = 210\n]", "---", "### Solve for (n) Step-by-Step", "1. Multiply both sides by 2 to eliminate the denominator:", "[\nn(n + 1) = 420\n]", "2. Expand the left-hand side:", "[\nn^2 + n = 420\n]", "3. Rearrange into standard quadratic form:", "[\nn^2 + n - 420 = 0\n]", "4. Apply the quadratic formula:\nThe quadratic equation (an^2 + bn + c = 0) has solutions:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 1), (b = 1), and (c = -420). Plugging in:", "[\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]", "5. Simplify the square root:", "[\n\sqrt{1681} = 41\n]", "So:", "[\nn = \frac{-1 \pm 41}{2}\n]", "6. Calculate both roots:", "- (n = \frac{-1 + 41}{2} = \frac{40}{2} = 20)\n- (n = \frac{-1 - 41}{2} = \frac{-42}{2} = -21)", "Since (n) must be a positive integer (because we’re summing positive integers), we discard (-21).", "---", "### Final Answer", "[\n\boxed{n = 20}\n]", "---", "### Why This Matters", "This problem demonstrates how algebraic methods transform word problems into solvable equations. Recognizing the pattern of summation allows quick identification of formulas like (\frac{n(n+1)}{2}), saving time in algebra, computer science, and even economics where cumulative sums are key.", "Next time you see a sum like 210, recall this formula—you’ve just unlocked a powerful shortcut!", "---", "Keywords: sum of first (n) integers, arithmetic series formula, solve for (n), quadratic equation, positive integers sum, arithmetic progression formula, math problem solution", "Meta Description:\nDiscover how to find (n) when the sum of the first (n) positive integers equals 210. Learn the formula, solve the equation step-by-step, and understand the math behind this common problem."]
