Question:** The sum of the first \( n \) natural numbers is 210. Find \( n \).
["How to Solve: The Sum of the First ( n ) Natural Numbers Equals 210. Find ( n )", "Mathematics is full of elegant formulas and patterns — and one of the most famous examples is the sum of the first ( n ) natural numbers. If you’ve ever wondered how to quickly determine which number ( n ) makes this sum equal to 210, you’re in the right place. This article walks you through the logic, the formula, and the calculation to solve the question:", "The Sum of the First ( n ) Natural Numbers Is 210. Find ( n ).", "### Understanding the Formula", "The sum of the first ( n ) natural numbers is given by the well-known arithmetic formula:", "[\nS = \frac{n(n + 1)}{2}\n]", "This formula represents the total when you add all integers from 1 to ( n ). We’re told that this total equals 210:", "[\n\frac{n(n + 1)}{2} = 210\n]", "### Step-by-Step Calculation", "1. Multiply both sides by 2 to eliminate the fraction:", "[\nn(n + 1) = 420\n]", "2. Expand and rearrange into a standard quadratic equation:", "[\nn^2 + n - 420 = 0\n]", "3. Solve the quadratic equation using the quadratic formula ( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), where ( a = 1 ), ( b = 1 ), and ( c = -420 ):", "[\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]", "4. Calculate the square root:", "[\n\sqrt{1681} = 41\n]", "5. Simplify both solutions:", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n\quad \ ext{and} \quad\nn = \frac{-1 - 41}{2} = -21 \quad \ ext{(discard, since ( n ) must be positive)}\n]", "### Final Answer", "So, the value of ( n ) that satisfies the equation is:", "[\n\boxed{20}\n]", "### Why This Formula Matters", "Understanding how to calculate the sum of the first ( n ) natural numbers empowers you not just to solve this specific problem, but also to approach sequences, series, and real-world counting problems with confidence. Moreover, the formula itself is a gateway to arithmetic progressions and has applications in physics, finance, and computer science.", "Bonus Tip: Once you know the formula ( S = \frac{n(n+1)}{2} ), you can invert it quickly — as shown above — to find ( n ) given ( S = 210 ), saving time over brute-force trial and error.", "---", "Summary:", "- Use the formula ( S = \frac{n(n+1)}{2} ) for the sum of the first ( n ) natural numbers.\n- Set ( S = 210 ), leading to the quadratic equation ( n(n+1) = 420 ).\n- Solve the equation to find ( n = 20 ).\n- This elegant solution reflects the beauty and utility of algebra in everyday math problems.", "---", "Keywords for SEO: sum of first ( n ) natural numbers formula, how to find ( n ) given sum 210, sum formula 210, solve ( n ) when ( 1 + 2 + ... + n = 210 ), arithmetic series sum solution, calculation steps sum formula.", "---", "References:\n- Arithmetic Series Sum Formula\n- Algebraic Solution of Quadratic Equations\n- Mathematical Education Resources", "---", "Start solving summation puzzles today — with this formula, you’ll unlock a timeless mathematical secret!"]
