If the sum of the first \(n\) natural numbers is 210, find \(n\).
["# If the Sum of the First (n) Natural Numbers Is 210, Find (n)", "Understanding the formula for the sum of the first (n) natural numbers can unlock solutions to many number puzzles — especially when the total equals a known value like 210. Whether you're a student learning arithmetic series or a curious learner, knowing how to solve for (n) when the sum is given can simplify problem-solving dramatically.", "## What Is the Sum of the First (n) Natural Numbers?", "The sum of the first (n) natural numbers follows a classic mathematical formula:", "[\nS = \frac{n(n + 1)}{2}\n]", "This formula arises from pairing numbers in sequences, making it one of the most useful tools in basic algebra. When you know that this sum equals 210, you can set up an equation and solve for (n).", "## Setting Up the Equation", "Given:\n[\n\frac{n(n + 1)}{2} = 210\n]", "Multiply both sides by 2 to eliminate the denominator:", "[\nn(n + 1) = 420\n]", "Now expand the left-hand side:", "[\nn^2 + n = 420\n]", "Bring all terms to one side to form a quadratic equation:", "[\nn^2 + n - 420 = 0\n]", "## Solving the Quadratic Equation", "Solve this quadratic using factoring, the quadratic formula, or completing the square. First, attempt factoring:", "We seek two integers that multiply to (-420) and add to (+1). Testing factors of 420, we find:", "[\n(n + 21)(n - 20) = 0\n]", "Set each factor equal to zero:", "[\nn + 21 = 0 \quad \Rightarrow \quad n = -21 \quad \ ext{(not valid, } n \ ext{ must be positive)}\n]\n[\nn - 20 = 0 \quad \Rightarrow \quad n = 20\n]", "Thus, (n = 20) is the valid solution.", "## Verifying the Solution", "Check that the sum of the first 20 natural numbers equals 210:", "[\n\frac{20 \ imes (20 + 1)}{2} = \frac{20 \ imes 21}{2} = \frac{420}{2} = 210\n]", "This confirms our answer.", "## Why This Knowledge Matters", "Knowing how to find (n) from the sum formula builds deep conceptual understanding. It reinforces algebraic reasoning, enhances problem-solving speed, and serves as a gateway to more advanced math concepts like series and calculus.", "Whether for homework, interviews, or self-study, this method is quick, reliable, and widely applicable. Remember: when the sum is 210, the number of terms is clearly (n = 20).", "---", "Key Takeaways:\n- The sum of the first (n) natural numbers is (\frac{n(n+1)}{2}).\n- Setting this equal to 210 gives a quadratic equation.\n- Solving step-by-step confirms (n = 20).\n- Always verify your solution to ensure accuracy.", "Use this formula confidently — it’s more than a formula; it’s a pattern every learner should master."]
