If the sum of the first \( n \) natural numbers equals 55, what is \( n \)?
["If the Sum of the First ( n ) Natural Numbers Equals 55, What Is ( n )?", "Have you ever wondered how mathematicians solve problems like “If the sum of the first ( n ) natural numbers equals 55, what is ( n )?” This classic question blends basic arithmetic with a well-known formula, making it both a fundamental math concept and a great exercise in problem-solving.", "### Understanding the Sum of the First ( n ) Natural Numbers", "The sum of the first ( n ) natural numbers—like 1 + 2 + 3 + … + ( n )—is given by the formula:", "[\nS = \frac{n(n + 1)}{2}\n]", "This formula dates back to the mathematician Carl Friedrich Gauss, who famously derived it as a child by pairing numbers symmetrically. Regardless, it’s one of the most important and frequently used formulas in elementary number theory.", "### Solving for ( n ) When the Sum is 55", "We are told that:", "[\n\frac{n(n + 1)}{2} = 55\n]", "To find ( n ), follow these steps:", "1. Multiply both sides by 2 to eliminate the denominator:\n[\nn(n + 1) = 110\n]", "2. Expand the left-hand side:\n[\nn^2 + n = 110\n]", "3. Rearrange into standard quadratic form:\n[\nn^2 + n - 110 = 0\n]", "4. Solve this quadratic equation using the quadratic formula:\n[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \ ext{where } a = 1, b = 1, c = -110\n]", "[\nn = \frac{-1 \pm \sqrt{1 + 440}}{2} = \frac{-1 \pm \sqrt{441}}{2} = \frac{-1 \pm 21}{2}\n]", "This gives two solutions:\n[\nn = \frac{20}{2} = 10 \quad \ ext{and} \quad n = \frac{-22}{2} = -11\n]", "Since ( n ) must be a positive natural number, we discard ( -11 ). Therefore:", "[\nn = 10\n]", "### Why This Matters", "Knowing that the sum of the first 10 natural numbers equals 55 proves invaluable in many areas—grading large datasets, solving pattern problems, or analyzing cumulative growth models. More importantly, this problem introduces one of the earliest algebraic insights: transforming word problems into solvable equations.", "### Final Answer", "If the sum of the first ( n ) natural numbers equals 55, then ( n = 10 ).", "---", "Keywords: sum of first ( n ) natural numbers, ( n ) where sum is 55, formula for triangular numbers, solve ( n(n+1)/2 = 55 ), natural numbers sum, math problem solving, quadratic equation in math, formula derivation Gauss", "Meta Description:\nWhen the sum of the first ( n ) natural numbers equals 55, what is ( n )? Learn how to solve this classic equation using the triangular number formula and discover math fundamentals that apply in real-world problems."]
