The sum of the first \(n\) positive integers is given by the formula \(S = \frac{n(n+1)}{2}\). What is the sum of the first 50 positive integers?

["# The Sum of the First (n) Positive Integers: Formula and Example for (n = 50)", "Mathematics is full of elegant patterns, and one of the most celebrated is the sum of the first (n) positive integers. This simple yet powerful formula, (S = \frac{n(n+1)}{2}), has fascinated students and mathematicians alike for centuries. In this article, we’ll explore the formula in depth, explain how it works, and walk through a practical example: finding the sum of the first 50 positive integers.", "## Understanding the Formula for the Sum of Integers", "The sum (S) of the first (n) positive integers—(1 + 2 + 3 + \cdots + n)—can be calculated using the well-known formula:", "[\nS = \frac{n(n+1)}{2}\n]", "This formula is derived from a classic story attributed to the mathematician Carl Friedrich Gauss: as a child, he quickly added the numbers from 1 to 100 by pairing them (1 + 100, 2 + 99, etc.), always finding the same sum, 101, and multiplying by 50 pairs. This insight leads directly to the general formula.", "Mathematically, this can be proven using induction, algebraic manipulation, or even geometric reasoning with triangular numbers. But for most practical purposes, memorizing and applying the formula is sufficient and efficient.", "## Why the Formula Works", "To understand why (S = \frac{n(n+1)}{2}) is correct, let’s rewrite the sum in reverse:", "[\nS = 1 + 2 + 3 + \cdots + (n-1) + n = n + (n-1) + (n-2) + \cdots + 2 + 1\n]", "Adding both expressions term-by-term:", "[\n2S = (1 + n) + (2 + (n-1)) + (3 + (n-2)) + \cdots + (n + 1) = n(n+1)\n]", "Thus, dividing both sides by 2 gives:", "[\nS = \frac{n(n+1)}{2}\n]", "This elegant logic explains the formula and confirms its accuracy.", "## Applying the Formula: Find the Sum of the First 50 Positive Integers", "Using the formula, we can quickly compute the sum of the first 50 positive integers by substituting (n = 50):", "[\nS = \frac{50(50 + 1)}{2} = \frac{50 \ imes 51}{2}\n]", "First calculate the product:", "[\n50 \ imes 51 = 2550\n]", "Then divide by 2:", "[\nS = \frac{2550}{2} = 1275\n]", "Therefore, the sum of the first 50 positive integers is 1275.", "## Real-World Applications of the Sum Formula", "While this formula appears simple, it has broad utility:", "- Computer Science: Algorithms that compute cumulative sums use this formula for performance optimization.\n- Finance: It helps calculate interest accumulation on simple interest over sequential time periods.\n- Daily Life: From counting tokens or steps to analyzing data sequences, this formula simplifies many counting tasks.", "## Summary", "- The sum of the first (n) positive integers is (S = \frac{n(n+1)}{2}).\n- This formula offers a fast way to compute sums without repetitive addition.\n- For (n = 50), the sum is (1275).\n- Understanding and applying such patterns strengthens mathematical intuition and problem-solving skills.", "Whether you're solving math problems, coding algorithms, or organizing data, mastering this foundational formula empowers you with a powerful tool for faster, accurate calculations.", "---", "Key Takeaway:\nUse (S = \frac{n(n+1)}{2}) to instantly compute the sum of the first (n) positive integers—perfect for (n = 50), yielding (S = 1275)."]