The sum of the first \( n \) natural numbers is 210. Find \( n \).
["Discover How to Find ( n ) When the Sum of the First ( n ) Natural Numbers Is 210", "Have you ever wondered how to determine the number ( n ) when the sum of the first ( n ) natural numbers equals 210? This classic problem combines basic number theory with a practical application of arithmetic formulas—perfect for students, math enthusiasts, and anyone curious about summation series. In this article, we’ll explore how to solve for ( n ), explain the underlying formula, and show step-by-step how to find that ( n = 20 ). Let’s dive in!", "---", "### Understanding the Formula: Sum of the First ( n ) Natural Numbers", "The sum of the first ( n ) natural numbers is given by the well-known formula:", "[\nS_n = \frac{n(n + 1)}{2}\n]", "Here, ( S_n ) represents the sum, and ( n ) is the number of terms. In our case, we know:", "[\n\frac{n(n + 1)}{2} = 210\n]", "Our goal is to solve this equation for ( n ).", "---", "### Step-by-Step Solution", "Start with the equation:", "[\n\frac{n(n + 1)}{2} = 210\n]", "Multiply both sides by 2 to eliminate the fraction:", "[\nn(n + 1) = 420\n]", "Expand the left-hand side:", "[\nn^2 + n = 420\n]", "Rearrange into a standard quadratic equation:", "[\nn^2 + n - 420 = 0\n]", "---", "### Solving the Quadratic Equation", "We now solve ( n^2 + n - 420 = 0 ) using factoring, the quadratic formula, or by recognizing integer solutions.", "Let’s try factoring. We look for two numbers that multiply to ( -420 ) and add to ( 1 ).", "After testing factor pairs, we find:", "[\n420 = 20 \ imes 21 \quad \ ext{and} \quad 21 - 20 = 1\n]", "So, the equation factors as:", "[\n(n + 21)(n - 20) = 0\n]", "Set each factor equal to zero:", "- ( n + 21 = 0 \Rightarrow n = -21 ) (not valid, since ( n ) must be positive)\n- ( n - 20 = 0 \Rightarrow n = 20 )", "---", "### Verifying the Solution", "Check that the sum of the first 20 natural numbers equals 210:", "[\nS_{20} = \frac{20 \ imes 21}{2} = \frac{420}{2} = 210\n]", "Confirmed! The calculation matches the given total.", "---", "### Real-World Application and Educational Value", "Understanding how to compute the sum of natural numbers and solve for ( n ) reinforces foundational algebra skills. This problem often appears in early statistics, competitive math, and introductory programming exercises. Knowing the formula and method helps students tackle more complex problems involving series, sequences, and iterative sums.", "---", "### Final Answer", "[\n\boxed{n = 20}\n]", "The first 20 natural numbers sum to 210. This illustrates the elegant simplicity of arithmetic series formulas and encourages deeper exploration into number theory and algebra.", "---", "Keywords: sum of first ( n ) natural numbers, formula derivation, solve ( n ) for 210, arithmetic series, quadratic equation, natural numbers sum, math problem solving, algebra basics.", "---", "Meta Description: Learn how to find ( n ) when the sum of the first ( n ) natural numbers equals 210. Discover the formula, step-by-step solution, and verify ( n = 20 ) with clear explanations and real-world applications."]
