The sum of the first n natural numbers is 210. Find n.
["The Sum of the First n Natural Numbers is 210: Find n in Seconds", "Ever wondered how to find the number n when you know the sum of the first n natural numbers equals 210? If you’re a student, math enthusiast, or problem-solver curious about arithmetic series, this guide breaks down the solution clearly and quickly.", "---", "### Understanding the Sum Formula", "The sum of the first n natural numbers is given by the well-known formula:", "[\nS = \frac{n(n + 1)}{2}\n]", "Where:\n- ( S ) = sum of the first n natural numbers\n- ( n ) = the last number in the sequence", "---", "### Setting Up the Equation", "We’re told that the sum equals 210:", "[\n\frac{n(n + 1)}{2} = 210\n]", "Multiply both sides by 2 to eliminate the fraction:", "[\nn(n + 1) = 420\n]", "---", "### Solving the Quadratic Equation", "Expand and rearrange:", "[\nn^2 + n - 420 = 0\n]", "Now solve this quadratic equation using factoring.", "We look for two numbers whose product is (-420) and sum is (1). These numbers are (21) and (-20):", "[\n(n + 21)(n - 20) = 0\n]", "So, the solutions are:", "[\nn = -21 \quad \ ext{or} \quad n = 20\n]", "Since n must be a positive natural number, we discard (-21).", "---", "### Verifying the Answer", "Let’s confirm:\nSum of first 20 natural numbers:", "[\n\frac{20 \ imes 21}{2} = \frac{420}{2} = 210\n]", "✅ The math checks out.", "---", "### Conclusion", "The natural number n for which the sum is 210 is:", "[\n\boxed{20}\n]", "This basic yet elegant formula is foundational in number theory and appears in many areas from classroom math to advanced algorithms. Mastering it unlocks stronger problem-solving skills—whether you're solving for n or exploring deeper into sequences and series.", "---", "Keywords: sum of first n natural numbers, find n, arithmetic series formula, n such that sum is 210, math problem solution, natural numbers sum 210", "Meta Description: Find the value of n when the sum of the first n natural numbers equals 210. Step-by-step solution using the formula ( \frac{n(n+1)}{2} = 210 ). Learn how to solve for n quickly and accurately."]
