Three Consecutive Odd Integers with a Sum of 51: A Comprehensive Guide

When dealing with mathematical problems involving consecutive odd integers, it's important to understand the underlying principles and techniques. In this article, we'll explore the problem where three consecutive odd integers sum up to 51. We will break down the solution step-by-step, discuss the methods used, and provide a thorough understanding of the concept.

Problem Statement

Find three consecutive odd integers whose sum is 51.

Method 1: Algebraic Representation

Let the first consecutive odd integer be x. Then, the next two consecutive odd integers can be represented as x 2 and x 4. The sum of these numbers is given as 51.

Mathematically, we can write:

x (x 2) (x 4) 51

Simplifying this equation, we get:

3x 6 51

Subtracting 6 from both sides:

3x 45

Dividing by 3:

x 15

Therefore, the three consecutive odd integers are:

15, 17, and 19

Method 2: Using the Average

The average of three consecutive odd integers is the middle number. Given that the sum of the three integers is 51, the average is:

51 / 3 17

The three consecutive odd integers surrounding 17 are 15, 17, and 19.

General Proof

To prove that 15, 17, and 19 are indeed the correct integers, we can substitute them back into the original equation:

15 17 19 51

This confirms our solution.

Using Algebraic Formulas

We can set up the equation using algebraic expressions:

x (x 2) (x 4) 51

Again, simplifying this equation:

3x 6 51

Subtracting 6:

3x 45

Dividing by 3:

x 15

The three consecutive odd integers are 15, 17, and 19.

Conclusion

In conclusion, the three consecutive odd integers that sum up to 51 are 15, 17, and 19. This problem can be solved through various methods, such as algebraic representation, using the average, or applying algebraic formulas. Understanding these methods will help in solving similar problems involving consecutive odd integers.