The Sum Of 3 Consecutive Integers

Decoding the Sum of 3 Consecutive Integers: A Simple Formula and its Applications

Meta Description: Discover the simple formula for calculating the sum of three consecutive integers, explore its mathematical proof, and understand its practical applications in various problem-solving scenarios. Learn how to easily solve these types of problems.

Finding the sum of three consecutive integers is a common mathematical problem that appears in various contexts, from basic algebra to more advanced mathematical concepts. While you can certainly add the numbers individually, understanding the underlying formula provides a much quicker and more efficient method. This article will break down the formula, provide a simple proof, and show you how to apply it.

Understanding the Formula

The sum of three consecutive integers can be expressed concisely using a simple formula. Let's represent the first integer as 'n'. The next two consecutive integers will then be 'n + 1' and 'n + 2'. Therefore, the sum of these three consecutive integers is:

n + (n + 1) + (n + 2) = 3n + 3

This simplifies to: 3(n + 1)

This means that the sum of any three consecutive integers is always three times the middle integer.

Mathematical Proof

We can easily prove this formula using the principles of algebra. Let's start with the sum of three consecutive integers:

n + (n + 1) + (n + 2)

Combining like terms, we get:

3n + 3

Factoring out a 3, we arrive at our simplified formula:

3(n + 1)

Applying the Formula: Real-World Examples

Let's look at a few examples to solidify our understanding:

• Example 1: Find the sum of 5, 6, and 7.

  • Using the formula, n = 5. Therefore, 3(5 + 1) = 3(6) = 18.

• Example 2: Find the sum of -2, -1, and 0.

  • Here, n = -2. Therefore, 3(-2 + 1) = 3(-1) = -3.

• Example 3: If the sum of three consecutive integers is 36, what are the integers?

  • We know that 3(n + 1) = 36. Dividing both sides by 3 gives us n + 1 = 12. Solving for n, we get n = 11. Therefore, the integers are 11, 12, and 13.

Beyond Three Consecutive Integers

While this article focuses on three consecutive integers, you can expand the concept to find the sum of any number of consecutive integers. The formula becomes more complex but follows a similar logic. For example, the sum of four consecutive integers would be 4n + 6, and the sum of five consecutive integers would be 5n + 10. Notice a pattern emerging? This is a great area for further exploration!

Conclusion

The formula for the sum of three consecutive integers – 3(n + 1) – provides a powerful and efficient tool for solving various mathematical problems. By understanding the formula and its proof, you can significantly improve your problem-solving skills and tackle more complex mathematical challenges with greater ease. Remember to always clearly define your variable 'n' as the first integer in the sequence. This simple yet elegant solution is a testament to the beauty and practicality of mathematics.