The difference between the squares of two consecutive integers is 31. Find the smaller integer.
["The Difference Between the Squares of Two Consecutive Integers Is 31: How to Find the Smaller Integer", "Understanding the mathematical relationship between consecutive integers is a fundamental concept in number theory and algebra. One intriguing question often posed is: What is the difference between the squares of two consecutive integers, and can that difference equal 31? If so, what is the smaller of the two integers?", "This article explores the mathematics behind this problem, solves for the smaller integer, and explains how to apply this insight in equation solving and algebra.", "---", "### What Is the Difference Between Squares of Consecutive Integers?", "Let’s begin with two consecutive integers. Suppose one integer is ( n ), and the next consecutive integer is ( n + 1 ).", "The square of the first integer is:\n[\nn^2\n]", "The square of the next integer is:\n[\n(n + 1)^2 = n^2 + 2n + 1\n]", "The difference between these two squares is:\n[\n(n + 1)^2 - n^2 = (n^2 + 2n + 1) - n^2 = 2n + 1\n]", "So, the difference between the squares of two consecutive integers simplifies to:\n[\n\boxed{2n + 1}\n]", "This means the difference increases by 2 with each increase in ( n ), starting at 1 when ( n = 0 ).", "---", "### When Is This Difference Equal to 31?", "We now ask:\nFor what integer ( n ) is the difference ( 2n + 1 = 31 )?", "Set up the equation:\n[\n2n + 1 = 31\n]", "Solve for ( n ):\n[\n2n = 31 - 1 = 30\n]\n[\nn = \frac{30}{2} = 15\n]", "---", "### Verification", "Check the squares of 15 and 16:\n[\n15^2 = 225, \quad 16^2 = 256\n]\n[\n256 - 225 = 31\n]", "The difference is indeed 31.", "---", "### Conclusion", "The difference between the squares of two consecutive integers is always of the form ( 2n + 1 ). When this difference equals 31, the smaller integer is:\n[\n\boxed{15}\n]", "This result is a perfect example of how algebraic expressions reveal hidden numerical patterns, making it valuable both for homework help and deeper mathematical exploration.", "---", "Keywords for SEO:\n- Difference between squares of consecutive integers\n- Solve 2n + 1 = 31\n- Find smaller integer given square difference\n- Consecutive integers algebra problem\n- Mathematical pattern in squares", "Meta Description:\nDiscover how the difference between squares of two consecutive integers follows the formula (2n + 1). Learn that when the difference equals 31, the smaller integer is 15 — a perfect example of algebra and number theory in action."]
