The sum of the first \( n \) positive integers is 210. What is \( n \)?
["The Sum of the First ( n ) Positive Integers Is 210. What Is ( n )?", "Have you ever wondered how to quickly calculate the sum of the first ( n ) positive integers? This classic problem appears often in math competitions, school homework, and simply curious math exploration. Today, we’re solving a specific question: If the sum of the first ( n ) positive integers equals 210, what is ( n )?", "### Understanding the Formula", "The sum of the first ( n ) positive integers is given by a well-known mathematical formula:", "[\nS = \frac{n(n + 1)}{2}\n]", "This formula, attributed to the ancient mathematician Carl Friedrich Gauss (who famously devised it as a child), allows us to compute the sum without adding each number one by one. Plugging in ( S = 210 ), we set up the equation:", "[\n\frac{n(n + 1)}{2} = 210\n]", "### Solving for ( n )", "Start by multiplying both sides by 2 to eliminate the denominator:", "[\nn(n + 1) = 420\n]", "This expands to a simple quadratic equation:", "[\nn^2 + n - 420 = 0\n]", "### Factoring the Quadratic Equation", "We aim to factor ( n^2 + n - 420 ). Look for two numbers that multiply to ( -420 ) and add to ( 1 ). After testing, we find:", "[\nn^2 + n - 420 = (n + 21)(n - 20)\n]", "Setting each factor equal to zero gives:", "[\nn + 21 = 0 \quad \ ext{or} \quad n - 20 = 0\n]", "So, ( n = -21 ) or ( n = 20 ). Since ( n ) must be a positive integer, we discard ( -21 ).", "### Conclusion", "The only valid solution is:", "[\nn = 20\n]", "This means the sum of the first 20 positive integers equals 210:", "[\n1 + 2 + 3 + \cdots + 20 = \frac{20 \ imes 21}{2} = 210\n]", "### Why This Formula Matters", "Understanding this formula helps in solving a wide range of problems involving arithmetic sequences, time complexity in algorithms, and even financial calculations involving repeated patterns. Remembering that ( 1 + 2 + \cdots + n = \frac{n(n+1)}{2} ) empowers quick, accurate computations—ideal for both exams and real-life problem-solving.", "So next time you’re asked, “The sum of the first ( n ) positive integers is 210. What is ( n )?”, confidently answer ( \mathbf{n = 20} ).", "---", "Keywords: sum of first ( n ) integers formula, arithmetic series sum, find ( n ) when sum is 210, solve ( \frac{n(n+1)}{2} = 210 ), math problem solution, Gauss sum formula", "Meta Description:\nDiscover how to find ( n ) when the sum of the first ( n ) positive integers equals 210 using the formula ( \frac{n(n+1)}{2} = 210 ). Learn step-by-step how ( n = 20 ). Ideal for students and math enthusiasts."]
