Set up the equation: n(n + 1)/2 = 210

["# Setting Up and Solving the Equation: n(n + 1)/2 = 210\nAn Effective Algebraic Approach to Find the Value of n", "---", "## Introduction", "Equations that describe patterns in sequences often appear in math competitions, daily problem-solving, and algorithm design. One common form is the triangular number equation, expressed as:", "$$\n\frac{n(n + 1)}{2} = 210\n$$", "This equation arises naturally in problems involving sums of consecutive integers. Solving it helps uncover the value of ( n ) such that the sum of the first ( n ) natural numbers equals 210. In this article, we’ll explore how to set up and solve this equation step-by-step, using algebra and key mathematical principles.", "---", "## Understanding the Equation", "The left-hand side, ( \frac{n(n + 1)}{2} ), represents the sum of the first ( n ) natural numbers—a concept that dates back to the ancient mathematician Carl Friedrich Gauss, who famously solved such sums efficiently.", "- The formula defines a triangular number, where ( n ) is a positive integer.\n- Given ( n ), the equation computes the total sum from 1 to ( n ).", "Our goal is to find the integer ( n ) satisfying:", "$$\n\frac{n(n + 1)}{2} = 210\n$$", "---", "## Step 1: Set Up the Equation Correctly", "To begin, rewrite the equation clearly:", "$$\n\frac{n(n + 1)}{2} = 210\n$$", "Multiply both sides by 2 to eliminate the fraction:", "$$\nn(n + 1) = 420\n$$", "Now expand the left-hand side:", "$$\nn^2 + n = 420\n$$", "Bring all terms to one side to form a quadratic equation:", "$$\nn^2 + n - 420 = 0\n$$", "This standard quadratic form ( an^2 + bn + c = 0 ) is now ready for solution.", "---", "## Step 2: Solve the Quadratic Equation", "We solve:", "$$\nn^2 + n - 420 = 0\n$$", "Use the quadratic formula:", "$$\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "Here, ( a = 1 ), ( b = 1 ), ( c = -420 ). Substitute:", "$$\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-420)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 1680}}{2} = \frac{-1 \pm \sqrt{1681}}{2}\n$$", "Note that:", "$$\n\sqrt{1681} = 41 \quad \ ext{(since } 41 \ imes 41 = 1681\ ext{)}\n$$", "So:", "$$\nn = \frac{-1 \pm 41}{2}\n$$", "This yields two solutions:", "- ( n = \frac{-1 + 41}{2} = \frac{40}{2} = 20 )\n- ( n = \frac{-1 - 41}{2} = \frac{-42}{2} = -21 )", "---", "## Step 3: Interpret the Solution", "Since ( n ) represents the upper limit of a sequence of natural numbers, it must be a non-negative integer. Thus, we discard ( n = -21 ) and accept:", "$$\n\boxed{n = 20}\n$$", "Verify by plugging back:", "$$\n\frac{20 \cdot 21}{2} = \frac{420}{2} = 210\n$$", "Confirmed!", "---", "## Practical Applications and Why It Matters", "Understanding how to set up and solve equations like ( \frac{n(n+1)}{2} = C ) is valuable in:", "- Competitive math: Recognizing patterns in sequences quickly.\n- Algorithms: Calculating cumulative sums efficiently.\n- Daily reasoning: Estimating summations or verifying progress.", "---", "## Summary", "Setting up the equation ( \frac{n(n + 1)}{2} = 210 ) allows bracketing the value of ( n ) using algebra. By transforming it into a quadratic equation and applying the quadratic formula, we find the exact integer solution ( n = 20 ) — the 20th triangular number equals 210. Mastering this process strengthens algebraic reasoning and problem-solving flexibility.", "---", "### Keyword-rich SEO tags for this article:\ntriangular number equation, set up n(n+1)/2 = 210, solve quadratic equation, sum of first n natural numbers, mathematical problem solving, algebraic methods, intermediate algebra, triangular numbers formula", "---", "Digitally optimize with internal links to related articles on quadratic formulas and arithmetic sequences. Use schema markup to highlight step-by-step solution for rich snippets."]