Question:** The sum of the first n natural numbers is given by the formula \( S = rac{n(n+1)}{2} \). What is the sum of the first 50 natural numbers?

["The Sum of the First n Natural Numbers: Formula, Calculation, and Example with n = 50", "Understanding the sum of the first n natural numbers is a foundational concept in mathematics, especially in arithmetic, algebra, and number theory. Whether you're solving math problems, preparing for exams, or simply curious about sequences, knowing how to compute this sum efficiently can save time and improve accuracy.", "### The Formula for the Sum of the First n Natural Numbers", "The sum of the first n natural numbers — that is, 1 + 2 + 3 + … + n — can be calculated using the well-known formula:", "[\nS = \frac{n(n + 1)}{2}\n]", "This elegant formula allows you to find the total instantly without adding each number individually — especially helpful when n is large.", "### Why Is This Formula Correct?", "The formula can be derived in multiple ways, including mathematical induction, algebraic reasoning, and even geometric visualization (such as arranging dots in triangles). For a quick explanation:\nIf you pair numbers from the start and end (e.g., 1 + 50, 2 + 49, 3 + 48), each pair sums to n + 1. Since there are n numbers, there are n/2 such pairs. Multiplying gives n × (n + 1)/2.", "### Applying the Formula: Sum of the First 50 Natural Numbers", "Now, let’s apply this formula to compute the sum of the first 50 natural numbers.", "Here, n = 50. Plug the value into the formula:", "[\nS = \frac{50(50 + 1)}{2} = \frac{50 \ imes 51}{2} = \frac{2550}{2} = 1275\n]", "So, the sum of the first 50 natural numbers is 1275.", "### Real-World Applications of This Formula", "Beyond classroom problems, this formula appears in:", "- Computer algorithms that calculate cumulative sums.\n- Spaced repetition systems in education technology.\n- Physics and engineering, when modeling observations or discrete changes over time.", "### Summary", "- The sum of the first n natural numbers is given by ( S = \frac{n(n + 1)}{2} ).\n- For n = 50, the sum is 1275.\n- This formula delivers fast, accurate results, making it essential for students, educators, and professionals alike.", "Understanding and applying this simple yet powerful formula equips you with a valuable mathematical tool — one that simplifies problems involving series, averages, and patterns in nature and data.", "---", "Try it Yourself: Next time you need the sum of 1 to 50, remember: it’s just 50 × 51 ÷ 2 = 1275. Quick, elegant, and effective!"]