The sum of the first \(n\) positive integers is 5050. Find \(n\).
["Finding the Value of ( n ) When the Sum of the First ( n ) Positive Integers Is 5050", "Have you ever wondered how mathematicians efficiently calculate the sum of a sequence like the first ( n ) positive integers? One fascinating example is when this sum equals 5,050:\n[\n1 + 2 + 3 + \cdots + n = 5050\n]\nIn this article, we’ll explore how to find the value of ( n ) using a well-known formula — and why this number is special in mathematics.", "---", "### The Formula for the Sum of the First ( n ) Positive Integers", "The sum of the first ( n ) positive integers follows a simple yet powerful formula:\n[\nS = \frac{n(n + 1)}{2}\n]\nThis formula, known as Gauss’s formula, allows us to compute the sum without adding each number individually.", "---", "### Setting Up the Equation", "We’re given:\n[\n\frac{n(n + 1)}{2} = 5050\n]", "Multiply both sides by 2 to eliminate the denominator:\n[\nn(n + 1) = 10100\n]", "Now expand the left side:\n[\nn^2 + n - 10100 = 0\n]", "This is a quadratic equation in standard form:\n[\nn^2 + n - 10100 = 0\n]", "---", "### Solving the Quadratic Equation", "To solve for ( n ), use the quadratic formula:\n[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nHere, ( a = 1 ), ( b = 1 ), ( c = -10100 ). Substitute in:\n[\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-10100)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 40400}}{2} = \frac{-1 \pm \sqrt{40401}}{2}\n]", "Calculate ( \sqrt{40401} ). Try squaring nearby numbers:\n[\n200^2 = 40000,\quad 201^2 = 40401\n]\nSo, ( \sqrt{40401} = 201 ).", "Now substitute back:\n[\nn = \frac{-1 + 201}{2} = \frac{200}{2} = 100\n]\n(We discard the negative root since ( n ) must be positive.)", "---", "### Verifying the Result", "Check that the sum of the first 100 positive integers equals 5050:\n[\n\frac{100 \ imes 101}{2} = 5050\n]\nThe calculation confirms the result.", "---", "### Why This Value Is Special", "The sum ( 1 + 2 + \cdots + 100 = 5050 ) is more than a number — it’s a landmark in number theory and classic mathematics. Known since antiquity, it illustrates how algebraic methods simplify arithmetic progressions. This formula, attributed to Carl Friedrich Gauss, remains foundational in teaching sequences, series, and the development of mathematical reasoning.", "---", "### Conclusion", "To find ( n ) such that the sum of the first ( n ) positive integers is 5,050, we used the formula ( \frac{n(n+1)}{2} = 5050 ), solved the resulting quadratic equation, and found:\n[\nn = 100\n]\nThis elegant result demonstrates the power of algebraic insight in solving sum-based problems.", "---", "Keywords: sum of first ( n ) integers, sum formula, Gauss, quadratic equation, find ( n ), arithmetic series, mathematics education.", "---", "### Further Reading", "- Learn how arithmetic series formulas are used in statistics.\n- Explore generalizations: What is the sum ( 1 + 2 + \cdots + n^2 )?\n- Study historical accounts of Gauss’s story as a child mathematician.", "---", "Try it yourself:\nUse ( n(n+1)/2 = 5050 ) to practice solving quadratics. This sum appears frequently in competitive math and classroom problems — knowing ( n = 100 ) helps build confidence!"]
