Set up the equation: \( rac{n(n + 1)}{2} = 5050 \).

["Setting Up the Equation: ( \dfrac{n(n + 1)}{2} = 5050 ) – A Step-by-Step Guide", "Understanding how to set up and solve equations involving triangular numbers is essential for math learners and problem solvers alike. One of the most well-known formulas in number theory is the formula for the ( n )-th triangular number. In this article, we’ll explore how to properly set up the equation:", "[\n\dfrac{n(n + 1)}{2} = 5050\n]", "This equation represents the sum of the first ( n ) natural numbers — a classic problem that has fascinated mathematicians for centuries.", "---", "### What Are Triangular Numbers?", "The ( n )-th triangular number, often denoted ( T_n ), is the sum of all integers from 1 to ( n ). The formula for this sum is:", "[\nT_n = \frac{n(n + 1)}{2}\n]", "These numbers get their name because they can be visualized as triangular patterns of dots. For example:", "- ( T_1 = 1 )\n- ( T_2 = 1 + 2 = 3 )\n- ( T_3 = 1 + 2 + 3 = 6 )\n- and so on.", "---", "### Why Set Up ( \dfrac{n(n + 1)}{2} = 5050 )?", "The number 5050 appears frequently in problems involving triangular numbers because it is a well-known triangular number:", "[\nT_{100} = \dfrac{100 \cdot 101}{2} = 5050\n]", "So, solving ( \dfrac{n(n + 1)}{2} = 5050 ) helps us determine which position ( n ) reaches this specific cumulative sum. This is a classic algebraic exercise that combines equation solving with real-world application.", "---", "### How to Set Up the Equation", "To formalize the concept into a solvable algebraic expression, observe the definition of triangular numbers. The sum of the first ( n ) integers equals 5050. Thus, we set up:", "[\n\dfrac{n(n + 1)}{2} = 5050\n]", "This equation equates the triangular number formula to the known value 5050, setting the stage for algebraic manipulation.", "---", "### Solving the Equation", "To solve ( \dfrac{n(n + 1)}{2} = 5050 ), follow these steps:", "1. Multiply both sides by 2 to eliminate the denominator:", "[\nn(n + 1) = 10100\n]", "2. Expand the left side:", "[\nn^2 + n = 10100\n]", "3. Rearrange into standard quadratic form:", "[\nn^2 + n - 10100 = 0\n]", "4. Apply the quadratic formula:", "[\nn = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 1 ), and ( c = -10100 ). Plugging in:", "[\nn = \dfrac{-1 \pm \sqrt{1^2 - 4(1)(-10100)}}{2(1)} = \dfrac{-1 \pm \sqrt{1 + 40400}}{2} = \dfrac{-1 \pm \sqrt{40401}}{2}\n]", "5. Calculate the square root:", "[\n\sqrt{40401} = 201\n]", "6. Simplify:", "[\nn = \dfrac{-1 + 201}{2} = \dfrac{200}{2} = 100 \quad \ ext{(discard the negative root as ( n ) must be positive)}\n]", "---", "### Conclusion", "The equation ( \dfrac{n(n + 1)}{2} = 5050 ) correctly models the sum of the first 100 natural numbers. Solving it gives:", "[\n\boxed{n = 100}\n]", "This elegant solution demonstrates how algebraic equations can capture fundamental mathematical truths. Whether you're studying number theory, preparing for math competitions, or teaching students, setting up the equation thoughtfully unlocks deeper understanding and problem-solving skills.", "Key takeaways:", "- Triangular numbers are sums of sequential integers.\n- The formula ( T_n = \dfrac{n(n+1)}{2} ) is central to solving such problems.\n- Setting up ( \dfrac{n(n + 1)}{2} = 5050 ) transforms a geometric idea into a solvable equation.\n- Quadratic formula enables finding ( n ) accurately.", "Mastering this setup empowers you to work confidently with triangular numbers and similar patterns in mathematics."]