Introduction to Consecutive Integers

Consecutive integers are a fascinating mathematical concept often appearing in algebraic problems. In this article, we will explore how to find three consecutive integers given a specific sum. This topic is not only helpful for academic purposes but also popular in search engine optimization (SEO) content. By providing detailed, step-by-step solutions and keywords, we aim to fulfill the SEO needs while ensuring the content is informative and engaging for readers.

Understanding the Problem: The Sum of Three Consecutive Integers

Let's start by defining the problem: the sum of three consecutive integers is 72. Consecutive integers are a sequence of numbers where each number is one more than the previous one. For instance, in the sequence 20, 21, 22, 20 is the first, 21 is the second, and 22 is the third.

Solution Method

Let the first consecutive number be n. The second consecutive number would be n 1, and the third consecutive number would be n 2. We can write the following linear equation to represent the sum of these three integers:

n (n 1) (n 2) 72

Step-by-Step Solution

Now, let's solve this equation for n:

Collect like terms on the left side: n n 1 n 2 72 3n 3 72 Subtract 3 from both sides to isolate the term with n: 3n 69 Divide both sides by 3 to completely isolate n: n 23

Therefore, the three consecutive integers are:

n 23 n 1 24 n 2 25

Checking the answer:

23 24 25 72

Verification confirms that the solution is correct.

Generalization and Application

The process can be generalized for any given sum. For a given sum of 72, the method holds true. However, it's important to note that the solution may vary for different sums. For example, if the sum is 63, the consecutive integers would be 20, 21, and 22. The same logic applies to finding the integers for sums of 90, which would be 29, 30, and 31. This can be proven using the same algebraic manipulation but substituting the values accordingly.

Conclusion: SEO Optimization

Understanding and solving problems involving consecutive integers can be a valuable SEO topic, especially in educational and academic fields. By including detailed step-by-step solutions and relevant keywords like 'consecutive integers,' 'linear equation,' and 'multiples of three,' we can effectively optimize the content for search engines. This guide not only provides clear, concise information but also encourages readers to delve into similar problems, enhancing engagement and backlinks.