The Sum Of Three Consecutive Integers Is

The Sum of Three Consecutive Integers: A Mathematical Exploration

Meta Description: Discover the simple formula and underlying logic behind finding the sum of three consecutive integers. This article provides a clear explanation, examples, and explores the broader mathematical concept.

Finding the sum of three consecutive integers is a fundamental concept in mathematics, useful not only for solving simple arithmetic problems but also for understanding more advanced algebraic principles. This article will break down the process, demonstrating the ease of calculation and the underlying mathematical reasoning. We'll explore how to find the sum, provide examples, and even touch upon how this concept extends to larger sequences of consecutive numbers.

Understanding Consecutive Integers

Consecutive integers are whole numbers that follow each other in order, without any gaps. For example, 1, 2, and 3 are consecutive integers, as are 10, 11, and 12, or -5, -4, and -3. The key is that each number is one greater than the preceding number.

Deriving the Formula

Let's represent the 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 these three expressions together:

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

Simplifying this expression, we get:

3n + 3

This simplifies further to:

3(n + 1)

This elegant formula reveals that the sum of three consecutive integers is always three times the middle integer. This is a powerful shortcut that eliminates the need for adding the three numbers individually.

Examples

Let's illustrate this with a few examples:

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

Using our formula, the middle integer is 6. Therefore, the sum is 3 * 6 = 18. (5 + 6 + 7 = 18)

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

The middle integer is -1. The sum is 3 * (-1) = -3. (-2 + -1 + 0 = -3)

• Example 3: Find the sum of 100, 101, and 102.

The middle integer is 101. The sum is 3 * 101 = 303. (100 + 101 + 102 = 303)

Extending the Concept

The principle of finding the sum of consecutive integers can be extended to any number of consecutive integers. While the formula changes for different sequences, the underlying concept of algebraic representation and simplification remains the same. For instance, the sum of four consecutive integers can be expressed as 4n + 6, where 'n' is the first integer.

Conclusion

The sum of three consecutive integers is a simple yet insightful mathematical concept. Understanding the formula, 3(n + 1), provides a quick and efficient method for calculating the sum, highlighting the elegance and power of algebraic representation. This understanding lays a strong foundation for exploring more complex mathematical problems involving sequences and series.