The Sum Of Three Consecutive Integers

The Sum of Three Consecutive Integers: A Simple Mathematical Exploration

Meta Description: Discover the fascinating pattern behind the sum of three consecutive integers and learn how to easily calculate it using a simple formula. This guide explores the underlying mathematics and provides practical examples.

Finding the sum of three consecutive integers is a seemingly simple mathematical problem, yet it reveals a beautiful and consistent pattern that can be expressed with a concise formula. Understanding this pattern isn't just about solving a specific type of problem; it's about developing a deeper understanding of mathematical sequences and algebraic reasoning. This article will explore the concept, derive the formula, and provide practical examples to solidify your understanding.

Understanding Consecutive Integers

First, let's define our terms. Consecutive integers are whole numbers that follow each other in order without any gaps. Examples include:

• 1, 2, 3
• 10, 11, 12
• -5, -4, -3

Notice that each number in the sequence is one greater than the previous number.

Deriving the Formula

Let's represent our three consecutive integers algebraically. We can use 'n' to represent the first integer. The next two consecutive integers will then be 'n + 1' and 'n + 2'.

To find the sum, we simply add them together:

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

Combining like terms, we get:

3n + 3

We can further simplify this by factoring out a 3:

3(n + 1)

This is our formula! The sum of three consecutive integers is always three times the middle integer.

Applying the Formula: Examples

Let's test our formula with a few examples:

Example 1: Find the sum of 4, 5, and 6.

Using our formula, where n = 4, the middle integer is 5. Therefore, the sum is 3 * 5 = 15. (4 + 5 + 6 = 15)

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

Here, n = -2, and the middle integer is -1. The sum is 3 * (-1) = -3. (-2 + -1 + 0 = -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, we get n + 1 = 12. Therefore, n = 11. The three consecutive integers are 11, 12, and 13. (11 + 12 + 13 = 36)

Extending the Concept

While this article focuses on three consecutive integers, the principles can be extended to find the sum of any number of consecutive integers. The pattern and the ability to represent numbers algebraically are key skills in more advanced mathematical concepts.

Conclusion

The seemingly simple problem of summing three consecutive integers offers a valuable lesson in algebraic manipulation and pattern recognition. By understanding the formula 3(n + 1), we can quickly and efficiently calculate the sum of any three consecutive integers, demonstrating the power of mathematical principles to simplify complex problems. This fundamental understanding lays a solid foundation for tackling more advanced mathematical challenges in the future.