The sum of the first \( n \) odd numbers is 625. What is \( n \)?
["# The Sum of the First ( n ) Odd Numbers Is 625 – What Is ( n )?", "P humming a familiar pattern in your mind, you might already know a powerful little secret about numbers: the sum of the first ( n ) odd numbers is a perfect square — specifically, ( n^2 ). Want to know how this works, and how to solve the riddle when that sum equals 625?", "## Why Is the Sum of the First ( n ) Odd Numbers Equal to ( n^2 )?", "The sequence of odd numbers begins: 1, 3, 5, 7, 9, … Each term follows the pattern:\nodd number ( k = 2k - 1 ) starting from ( k = 1 ).", "Let’s add the first few odd numbers:\n- Sum of first 1 odd number: ( 1 = 1^2 )\n- Sum of first 2: ( 1 + 3 = 4 = 2^2 )\n- Sum of first 3: ( 1 + 3 + 5 = 9 = 3^2 )\n- Sum of first 4: ( 1 + 3 + 5 + 7 = 16 = 4^2 )\n- … and so on.", "Mathematically, this happens because the sum of the first ( n ) odd numbers is:\n[ 1 + 3 + 5 + \cdots + (2n - 1) = n^2 ]", "So, if this total sum equals 625, we’re looking for:\n[ n^2 = 625 ]", "## How to Find ( n ) When the Sum Is 625?", "Solve the equation simply:\n[ n = \sqrt{625} = 25 ]", "✅ Answer: ( n = 25 )", "That means the 25th odd number (which is ( 2 \ imes 25 - 1 = 49 )) completes the sum that equals ( 25^2 = 625 ).", "## Quick Verification", "Let’s add the first 25 odd numbers using the formula:\n[ \ ext{Sum} = n^2 = 25^2 = 625 ] ✔️", "Alternatively, summing from 1 to 49 in steps of 2 confirms:\n1 + 3 + 5 + … + 49 = 625 (verified via arithmetic series or spreadsheet).", "## Why This Formula Matters", "This simple relationship isn’t just a neat trick — it’s foundational in number theory and geometry. The fact that odd numbers form perfect squares connects deeply with patterns in art, puzzles like the Pythagorean theorem, and even quantum physics.", "---", "Bottom line: If the sum of the first ( n ) odd numbers is 625, then ( n = 25 ). This elegant result shows how mathematics reveals surprising harmony behind simple sequences.", "---", "Keywords: sum of first ( n ) odd numbers, ( n^2 = 625), arithmetic series odd numbers, mathematical pattern odd numbers, perfect square sequence, ( n ) where sum = 625, identity sum of odd numbers."]
