Find Three Consecutive Even Integers Whose Sum Is 108.

Find Three Consecutive Even Integers Whose Sum is 108: A Comprehensive Guide

Finding three consecutive even integers that add up to 108 might seem like a simple math problem, but it's a great example to illustrate several important mathematical concepts and problem-solving strategies. This article will guide you through different approaches to solving this problem, expanding on the core concepts and providing context for broader application. We'll explore algebraic solutions, intuitive methods, and even delve into the underlying mathematical principles. This isn't just about finding the answer; it's about understanding how we find it, and why it works.

Understanding the Problem: Defining Consecutive Even Integers

Before we dive into the solutions, let's clearly define the terms. "Consecutive even integers" means a sequence of even numbers where each number is two greater than the previous one. For example, 2, 4, 6 are consecutive even integers. The problem asks us to find three such numbers that add up to 108.

Keywords: consecutive even integers, sum, algebra, equation, problem-solving, mathematics

Method 1: The Algebraic Approach

This is the most common and generally preferred method for solving this type of problem. We use variables to represent the unknown integers and create an equation based on the given information.

Setting up the Equation

Let's represent the three consecutive even integers as:

• x: The first even integer
• x + 2: The second even integer (two greater than the first)
• x + 4: The third even integer (four greater than the first)

The problem states that the sum of these three integers is 108. Therefore, we can write the equation:

x + (x + 2) + (x + 4) = 108

Solving the Equation

Now, let's solve this equation for 'x':

1. Combine like terms: 3x + 6 = 108
2. Subtract 6 from both sides: 3x = 102
3. Divide both sides by 3: x = 34

Therefore, the first even integer (x) is 34.

Finding the Other Integers

Now that we know x = 34, we can find the other two consecutive even integers:

• x + 2 = 34 + 2 = 36
• x + 4 = 34 + 4 = 38

Verifying the Solution

Let's check if our solution is correct by adding the three integers:

34 + 36 + 38 = 108

The sum is indeed 108, so our solution is correct. The three consecutive even integers are 34, 36, and 38.

Method 2: The Intuitive Approach

While the algebraic method is precise, an intuitive approach can be helpful for understanding the problem's core logic, particularly for smaller numbers.

Dividing by Three

Since we're looking for three numbers that add up to 108, a good starting point is to divide 108 by 3:

108 / 3 = 36

This gives us the average of the three integers. Because we are dealing with consecutive even numbers, the middle number is the average.

Finding the Other Integers

Knowing that the middle number is 36, we can easily find the consecutive even integers before and after it:

• Before 36: 34
• After 36: 38

Therefore, the three consecutive even integers are 34, 36, and 38. This method offers a quicker solution but relies on the inherent properties of consecutive even integers and might not be as easily adaptable to more complex problems.

Method 3: A More General Approach using Arithmetic Sequences

This method leverages the concept of arithmetic sequences. Consecutive even integers form an arithmetic sequence with a common difference of 2. The sum of an arithmetic sequence can be calculated using the formula:

S = n/2 * [2a + (n-1)d]

Where:

• S is the sum of the sequence
• n is the number of terms (in this case, 3)
• a is the first term (which is what we want to find)
• d is the common difference (which is 2 for consecutive even integers)

Applying the Formula

Let's plug in the values we know:

108 = 3/2 * [2a + (3-1)2]

Solving for 'a'

1. Multiply both sides by 2/3: 72 = 2a + 4
2. Subtract 4 from both sides: 68 = 2a
3. Divide both sides by 2: a = 34

This gives us the first term, 'a', which is 34. The consecutive even integers are then 34, 36, and 38. This method demonstrates a more generalized approach applicable to finding consecutive even or odd integers summing to any given number.

Expanding the Concepts: Applications and Further Exploration

The problem of finding consecutive even integers with a specific sum is a foundational problem that connects to several broader mathematical concepts:

• Arithmetic Progressions: Understanding arithmetic sequences and their properties is crucial in various mathematical fields, including number theory and calculus.
• Equation Solving: The algebraic approach reinforces fundamental equation-solving skills, essential for tackling more complex problems in algebra and other branches of mathematics.
• Problem-Solving Strategies: The different methods presented highlight various problem-solving approaches – systematic, intuitive, and formula-based – each useful in different contexts.
• Real-World Applications: While this specific problem might seem abstract, similar concepts are applied in real-world scenarios involving evenly spaced data points, resource allocation, or financial calculations.

Beyond the Numbers: Developing Problem-Solving Skills

This simple problem serves as an excellent exercise in developing crucial problem-solving skills:

• Breaking Down Complex Problems: The problem is broken down into smaller, manageable steps. This approach is invaluable for tackling larger, more complicated problems.
• Identifying Patterns and Relationships: Recognizing the pattern of consecutive even integers and the relationship between their sum and their average is key to efficient problem-solving.
• Verification and Validation: Checking the solution confirms its accuracy. This habit ensures reliable results and is crucial in all aspects of problem-solving.
• Choosing the Right Method: Understanding the strengths and weaknesses of different methods allows you to select the most efficient approach for specific problems.

Conclusion: Mastering the Fundamentals

Finding three consecutive even integers that sum to 108 is more than just a mathematical puzzle; it's a journey into fundamental mathematical concepts and effective problem-solving techniques. Whether you prefer the elegance of algebra or the intuitive nature of the average method, understanding the underlying principles empowers you to tackle similar problems with confidence and efficiency. This exercise builds a strong foundation for tackling more complex mathematical challenges in the future. The ability to approach a problem from multiple angles and verify solutions is a valuable skill applicable far beyond the realm of mathematics.