If the sum of the first \( n \) positive integers is 210, find \( n \).
["Unlocking the Mystery: How to Find ( n ) When the Sum of the First ( n ) Positive Integers Is 210", "Mathematics is full of elegant patterns, and one of the most fascinating is the sum of the first ( n ) positive integers. If you’ve ever wondered how to find the number ( n ) when the total sum equals 210, this article perfectly explains how to solve the problem using a simple formula.", "The sum of the first ( n ) positive integers follows a well-known formula:", "[\nS = \frac{n(n + 1)}{2}\n]", "This formula comes from the arithmetic series sum formula, where you add the first ( n ) numbers: ( 1 + 2 + 3 + \dots + n ). Given that this sum equals 210, we set up the equation:", "[\n\frac{n(n + 1)}{2} = 210\n]", "Multiplying both sides by 2 eliminates the fraction:", "[\nn(n + 1) = 420\n]", "Now we have a quadratic equation:", "[\nn^2 + n - 420 = 0\n]", "Solving this quadratic is straightforward. We look for two numbers whose product is ( -420 ) and sum is ( 1 ). Factoring, we find:", "[\n(n + 21)(n - 20) = 0\n]", "This gives two solutions: ( n = -21 ) and ( n = 20 ). Since ( n ) must be a positive integer, we discard ( -21 ) and accept:", "[\nn = 20\n]", "Thus, the sum of the first 20 positive integers is:", "[\n\frac{20 \cdot 21}{2} = 210\n]", "Why This Formula Matters", "Understanding how to compute the sum of the first ( n ) integers is essential in many areas of math—from simple arithmetic problems to advanced calculus and algorithms. Recognizing patterns like this builds problem-solving intuition and lays the foundation for more complex series and summations.", "Key Takeaways", "- The sum of the first ( n ) positive integers is given by ( \frac{n(n + 1)}{2} ).\n- Setting this equal to 210 leads to a quadratic equation with a clear positive integer solution.\n- Verification confirms that ( n = 20 ) satisfies the condition perfectly.", "Whether you're a student tackling homework or a curious learner exploring math, this principle offers a quick and satisfying way to unlock a classic mathematical identity.", "Start solving more integer sum problems today—and discover how simple equations can reveal powerful truths about numbers!", "---", "Keywords: solve for ( n ), sum of first ( n ) integers 210, find ( n ) such that ( 1 + 2 + \dots + n = 210 ), arithmetic series formula, quadratic equation integer solution"]
