The sum of three consecutive even integers is 126. What is the largest integer?
["The Sum of Three Consecutive Even Integers Is 126: What Is the Largest Integer?", "Finding three consecutive even integers whose sum equals 126 is a classic math challenge that reveals the elegant structure of numbers. Not only is it a great exercise in algebra, but it also helps reinforce logic and pattern recognition—skills essential for students and math enthusiasts alike. In this article, we’ll break down how to solve this problem step by step and identify the largest of the three even integers.", "### Understanding Consecutive Even Integers", "Consecutive even integers are numbers that follow one after another in order, with a difference of 2 between each. For example, 10, 12, and 14 are consecutive even integers. They can be expressed using a common variable to reflect their relationship.", "Let the middle even integer be represented as:\nx\nThen the three consecutive even integers are:\n- First: x – 2\n- Second (middle): x\n- Third: x + 2", "### Setting Up the Equation", "The problem states that their sum is 126. So we write:", "[\n(x - 2) + x + (x + 2) = 126\n]", "Simplify the left-hand side:\n[\nx - 2 + x + x + 2 = 126\n]", "Combine like terms:\n[\n3x = 126\n]", "### Solving for x", "Divide both sides by 3:\n[\nx = \frac{126}{3} = 42\n]", "So the middle integer is 42. Now determine the three consecutive even integers:\n- First: (42 - 2 = 40)\n- Second: (42)\n- Third: (42 + 2 = 44)", "### Identifying the Largest Integer", "The largest of the three integers is 44.", "### Verifying the Solution", "Let’s double-check:\n[\n40 + 42 + 44 = 126\n]\nYes, the sum is correct.", "### Why This Method Works", "Using a variable for the middle number simplifies the arithmetic and avoids confusion that might come from labeling the first integer. Since the total sum is fixed, symmetric distribution around the middle makes solving straightforward and reliable.", "### Real-World Applications", "Understanding sums of consecutive even integers helps in:\n- Pattern recognition in sequences\n- Solving word problems in school and standardized tests\n- Applying algebra to real-life scenarios such as budgeting or planning evenly spaced events", "### Summary", "To solve “the sum of three consecutive even integers is 126. What is the largest integer?” follow these steps:\n1. Represent the middle number as (x).\n2. Express the three numbers as (x-2), (x), (x+2).\n3. Set up the equation and solve for (x).\n4. Identify the largest integer as (x + 2).", "In this case, the largest integer is 44.", "---", "Understanding these foundational math concepts empowers you not just to solve problems like this, but to approach numbers with confidence and clarity every day."]
