The sum of the first \( n \) natural numbers is 210. What is \( n \)?
["The Sum of the First ( n ) Natural Numbers Is 210—Here’s How to Find ( n )", "Have you ever wondered how to find the sum of the first ( n ) natural numbers when you already know the total? If the sum equals 210, what is ( n )? Solving this problem explores a classic formula in mathematics and improves your problem-solving skills with arithmetic sequences.", "### The Formula for the Sum of the First ( n ) Natural Numbers", "The sum of the first ( n ) natural numbers is given by the arithmetic series formula:", "[\nS_n = \frac{n(n + 1)}{2}\n]", "This formula allows us to calculate the total sum quickly without adding all numbers one by one.", "### Setting Up the Equation", "We’re told:", "[\nS_n = 210\n]", "Using the formula, we write:", "[\n\frac{n(n + 1)}{2} = 210\n]", "### Solving for ( n )", "Multiply both sides by 2 to eliminate the fraction:", "[\nn(n + 1) = 420\n]", "This becomes a quadratic equation:", "[\nn^2 + n - 420 = 0\n]", "Now solve this quadratic equation using factoring. We look for two numbers whose product is (-420) and sum is (1). These numbers are (21) and (-20):", "[\n(n + 21)(n - 20) = 0\n]", "Setting each factor to zero gives:", "[\nn + 21 = 0 \quad \ ext{or} \quad n - 20 = 0\n]", "[\nn = -21 \quad \ ext{or} \quad n = 20\n]", "Since ( n ) must be a positive integer (a natural number), we discard (-21).", "### Conclusion", "Therefore,", "[\nn = 20\n]", "### Verification", "Check by plugging ( n = 20 ) into the sum formula:", "[\nS_{20} = \frac{20(20 + 1)}{2} = \frac{20 \ imes 21}{2} = \frac{420}{2} = 210\n]", "The sum matches perfectly.", "---", "Why This Matters:\nUnderstanding how to derive ( n ) from the sum uses mathematical reasoning and algebra. It appears in real-life scenarios, from organizing groups to solving planning problems efficiently.", "So the next time someone tells you the sum of the first ( n ) natural numbers is 210, you’ll know exactly that ( n = 20 )—and how to get there.", "---", "Keywords: sum of first ( n ) natural numbers, find ( n ), sum formula, arithmetic series, solve quadratic equation, ( n = 20 ), mathematics education", "Meta Description:\nDiscover how to find ( n ) when the sum of the first ( n ) natural numbers is 210 using the arithmetic series formula. Step-by-step solution with verification."]
