The sum of the first \(n\) integers is \(\frac{n(n+1)}{2} = 210\).

["The Sum of the First (n) Integers Equals 210: How to Solve This Classic Math Problem", "Finding the sum of the first (n) positive integers is a timeless math challenge—perfect for beginners and a staple in arithmetic and algebra learning. If you’ve come across the equation:", "[\n\frac{n(n+1)}{2} = 210\n]", "you're on the right track to understanding how to solve for (n) step by step. In this article, we’ll explore how to solve this equation, why it works, and how to apply it in real-world scenarios. Let’s dive in!", "---", "### What Is the Sum of the First (n) Integers?", "The sum of the first (n) positive integers means adding:", "[\n1 + 2 + 3 + \cdots + n\n]", "Mathematically, this sum has a neat closed-form formula:", "[\nS_n = \frac{n(n+1)}{2}\n]", "This formula allows you to compute the sum instantly without adding each number one by one.", "---", "### Solving (\frac{n(n+1)}{2} = 210)", "We start with the equation:", "[\n\frac{n(n+1)}{2} = 210\n]", "Multiply both sides by 2 to eliminate the denominator:", "[\nn(n+1) = 420\n]", "Now rearrange into standard quadratic form:", "[\nn^2 + n - 420 = 0\n]", "This is a quadratic equation of the form (an^2 + bn + c = 0), where (a = 1), (b = 1), and (c = -420). We can solve it using the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute 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 get:", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20 \quad \ ext{(discard the negative root, as (n) must be positive)}\n]", "---", "### Verifying the Answer", "Let’s confirm that the sum of the first 20 integers equals 210:", "[\n\frac{20(20+1)}{2} = \frac{20 \ imes 21}{2} = \frac{420}{2} = 210\n]", "✅ Confirmed! The solution is correct.", "---", "### Why This Formula Matters", "Understanding how to derive and solve this equation improves problem-solving skills in algebra and introduces key concepts such as:", "- Quadratic equations\n- Summation formulas\n- Pattern recognition in number sequences", "It’s also essential for challenges like finding missing numbers in sequences, calculating series sums, or preparing for math competitions.", "---", "### Real-World Applications", "Besides classrooms and competitions, this formula appears in:", "- Page numbering in books – to find total pages\n- Workstation task batching – calculating total tasks in linear increments\n- Physics and engineering – summing discrete force or velocity data\n- Computer algorithms – optimizing loop-based summations", "---", "### Final Thoughts", "Solving (\frac{n(n+1)}{2} = 210) may seem a small puzzle, but it exemplifies a foundational math ability—transforming a sum into an equation and mastering its solution. Whether you're a student, teacher, or curious learner, grasping this method unlocks deeper insights into algebra and problem-solving.", "Try it yourself: next time you see a sequence of consecutive numbers, ask—what’s their total? You now have the formula and the skill to compute it!", "---", "Keywords for SEO: sum of first (n) integers, formula for sum of integers, solve (\frac{n(n+1)}{2} = 210), quadratic formula application, arithmetic series proof, math lesson for students, algebra quick challenge, sum from 1 to (n), discrete mathematics, closed-form solution, math problem solving.", "---", "By mastering such equations, you’re not just solving a problem—you’re building a toolkit for logical thinking and advanced math. Keep practicing, and turn every sequence into a solved mystery!"]