The sum of the first \( n \) integers is given by \( \frac{n(n+1)}{2} = 210 \).
["# The Sum of the First ( n ) Integers Revealed: Solving ( \frac{n(n+1)}{2} = 210 )", "Understanding the sum of the first ( n ) positive integers is a foundational concept in mathematics that appears across various disciplines, from basic arithmetic to advanced algebra and computer science. One common problem encountered is solving the equation:", "[\n\frac{n(n+1)}{2} = 210\n]", "In this article, we’ll explore how to solve this equation step-by-step, explain its significance, and highlight practical applications and learning opportunities.", "---", "## What Is the Sum of the First ( n ) Integers?", "The sum of the first ( n ) natural numbers is given by the formula:", "[\nS = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\n]", "This formula, attributed to the legendary mathematician Carl Friedrich Gauss, provides a concise way to compute the total without manually adding each number.", "---", "## Solving ( \frac{n(n+1)}{2} = 210 )", "### Step 1: Multiply both sides by 2", "To eliminate the fraction, multiply both sides of the equation by 2:", "[\nn(n + 1) = 420\n]", "### Step 2: Expand and rearrange into a standard quadratic", "[\nn^2 + n - 420 = 0\n]", "This is a quadratic equation in the form ( an^2 + bn + c = 0 ), where ( a = 1 ), ( b = 1 ), and ( c = -420 ).", "### Step 3: Solve the quadratic equation", "We can use the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute the values:", "[\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]", "Since ( \sqrt{1681} = 41 ), we have:", "[\nn = \frac{-1 + 41}{2} = \frac{40}{2} = 20 \quad \ ext{or} \quad n = \frac{-1 - 41}{2} = \frac{-42}{2} = -21\n]", "---", "## Choosing the Valid Solution", "Since ( n ) must be a positive integer (representing a count of numbers), we discard ( n = -21 ). Thus:", "[\nn = 20\n]", "### Verification:", "[\n\frac{20(20 + 1)}{2} = \frac{20 \ imes 21}{2} = \frac{420}{2} = 210\n]", "This confirms the solution is accurate.", "---", "## Why This Formula Matters", "Understanding and applying the formula ( \frac{n(n+1)}{2} = 210 ) is more than an algebraic exercise—it helps develop problem-solving and pattern recognition skills. This specific instance arises in:", "- School math curricula: As an introduction to arithmetic series.\n- Programming challenges: Finding a number ( n ) such that the triangular number equals 210.\n- Mean calculations: Showing that the average of numbers 1 through 20 equals 10.5, confirming ( \frac{1 + 20}{2} = 10.5 ) and total sum 210.", "---", "## Summary", "- The sum of the first ( n ) integers is ( \frac{n(n+1)}{2} ).\n- Solving ( \frac{n(n+1)}{2} = 210 ) yields ( n = 20 ).\n- This formula builds intuition about number patterns, sequences, and real-world summations.", "---", "## Related Topics to Explore", "- Deriving the formula for the sum of the first ( n ) integers algebraically.\n- Applications of triangular numbers in geometry and combinatorics.\n- Using programming to solve or visualize integer sum problems.", "---", "## Key Takeaways", "- Always verify solutions in context—especially when solving for natural numbers.\n- The formula ( \frac{n(n+1)}{2} ) is a power tool in discrete mathematics.\n- Practicing problems like “sum equals 210” sharpens analytical thinking.", "---", "Want toMaster arithmetic series or explore triangular numbers further? Visit mathresources.com/triangular-numbers for interactive tools, quizzes, and deeper insights.", "---", "🔍 Blog SEO Keywords: sum of first n integers, triangular numbers formula, solve ∑1 to n = 210, arithmetic series sum, solving algebraic equations, math problem solving, n(n+1)/2 explained, natural numbers summation, integer series applications, Gaussian sum formula."]
