Setting this equal to 55: \( \frac{n(n+1)}{2} = 55 \).

["# Solving ( \frac{n(n+1)}{2} = 55 ): A Step-by-Step Guide to This Classic Equation", "When faced with the equation ( \frac{n(n+1)}{2} = 55 ), many students, math enthusiasts, and problem solvers seek to discover the value of ( n )—the key to unlocking a well-known triangular number. This formula represents the sum of the first ( n ) natural numbers, and in this case, it equals 55.", "In this article, we’ll explore how to solve ( \frac{n(n+1)}{2} = 55 ) step-by-step, why this equation arises in number theory, and how to apply this method to similar problems.", "---", "## Understanding the Equation", "The expression ( \frac{n(n+1)}{2} ) is the formula for the sum of the first ( n ) natural numbers:", "[\nS = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\n]", "Setting this equal to 55 gives us the equation:", "[\n\frac{n(n+1)}{2} = 55\n]", "Multiplying both sides by 2 to eliminate the denominator:", "[\nn(n+1) = 110\n]", "This leads to a simple quadratic equation:", "[\nn^2 + n - 110 = 0\n]", "---", "## Step 1: Solve the Quadratic Equation", "We solve the quadratic using factoring, completing the square, or the quadratic formula. In this case, factoring works cleanly.", "[\nn^2 + n - 110 = 0\n]", "We look for two numbers that multiply to (-110) and add to (1). These numbers are (11) and (-10):", "[\n(n + 11)(n - 10) = 0\n]", "Setting each factor to zero:", "- (n + 11 = 0 \Rightarrow n = -11) (discarded, since (n) must be a positive integer)\n- (n - 10 = 0 \Rightarrow n = 10)", "---", "## Step 2: Verify the Solution", "Substitute (n = 10) back into the original equation:", "[\n\frac{10 \cdot (10 + 1)}{2} = \frac{10 \cdot 11}{2} = \frac{110}{2} = 55\n]", "✓ Confirmed! The solution satisfies the equation.", "---", "## Why This Equation Matters", "Triangular numbers like 55 (which are the sums (1 + 2 + \cdots + n)) appear in various mathematical contexts—from combinatorics and number theory to algorithmic complexity and array sum problems.", "Solving ( \frac{n(n+1)}{2} = 55 ) demonstrates how quadratic equations model real-world counting and aggregation problems, and solving them builds foundational algebra skills needed for higher-level math.", "---", "## Practical Tips for Solving Similar Problems", "- Recognize the pattern: Familiarize yourself with triangular numbers, defined as ( T_n = \frac{n(n+1)}{2} ).\n- Rewrite the equation: Always multiply both sides by the denominator to eliminate fractions.\n- Factor carefully: When factoring quadratics, look for pairs of factors of (2c) (where (c) is the constant term) that add up to the coefficient of (n).\n- Verify your answer: Substitute your solution back into the original equation.", "---", "## Conclusion", "Setting ( \frac{n(n+1)}{2} = 55 ) and solving for ( n ) is a straightforward yet powerful exercise in algebra. The solution, ( n = 10 ), confirms that the sum of the first 10 natural numbers equals 55. This problem not only illustrates the use of quadratic equations but also connects to deeper mathematical themes involving sequences and sums.", "Whether you’re a student mastering algebra, a parent helping with homework, or someone exploring mathematical patterns, understanding how to solve this equation opens the door to solving countless related problems.", "---", "Keywords:\nTriangular number 55, solve ( \frac{n(n+1)}{2} = 55 ), algebra equation steps, sum of first ( n ) numbers, quadratic equation solution, math problem solving, n(n+1)/2 = 55, triangular number calculation", "Meta Description:\nLearn how to solve ( \frac{n(n+1)}{2} = 55 ) step-by-step, discover its connection to triangular numbers, and understand application in algebra and number theory. Perfect for math students and enthusiasts."]