Find Three Consecutive Odd Integers Whose Sum Is 117

Finding Three Consecutive Odd Integers that Sum to 117: A Comprehensive Guide

This article explores the mathematical problem of finding three consecutive odd integers that add up to 117. We'll delve into the problem-solving process step-by-step, explaining the underlying mathematical principles and providing a clear, concise solution. Understanding this problem not only helps in solving similar algebraic equations but also strengthens your fundamental understanding of number systems and algebraic manipulation. This guide is designed for students of all levels, from those just beginning algebra to those seeking to refine their problem-solving skills.

Understanding the Problem: Consecutive Odd Integers

Before diving into the solution, let's define the terms. "Consecutive odd integers" means a sequence of odd numbers following each other directly. For example, 1, 3, 5; 11, 13, 15; and 99, 101, 103 are all sets of consecutive odd integers. The key characteristic is that each number is two more than the preceding number.

Our goal is to find three such numbers that add up to 117. We'll use algebraic methods to solve this efficiently.

The Algebraic Approach: Setting up the Equation

To solve this problem elegantly, we can use algebra. Let's represent the three consecutive odd integers using variables:

Let 'x' represent the first odd integer.
Since the integers are consecutive and odd, the second integer will be 'x + 2'.
Similarly, the third integer will be 'x + 4'.

Now, we know that the sum of these three integers is 117. We can express this as an algebraic equation:

x + (x + 2) + (x + 4) = 117

Solving the Equation: Step-by-Step

Combine like terms: The equation simplifies to:

3x + 6 = 117

Isolate the variable: Subtract 6 from both sides of the equation:

3x = 111

Solve for x: Divide both sides by 3:

x = 37

Therefore, the first odd integer (x) is 37.

Finding the Other Two Integers

Now that we know the value of x, we can easily find the other two consecutive odd integers:

Second integer: x + 2 = 37 + 2 = 39
Third integer: x + 4 = 37 + 4 = 41

So, the three consecutive odd integers are 37, 39, and 41.

Verification: Checking the Solution

Let's verify our solution by adding the three integers together:

37 + 39 + 41 = 117

Our solution is correct! The sum of the three consecutive odd integers, 37, 39, and 41, is indeed 117.

A Deeper Dive: Generalizing the Solution

This problem can be generalized to find any set of consecutive odd integers that sum to a particular value. Let's say we want to find n consecutive odd integers whose sum is S. We can use a similar approach:

Let the first odd integer be x. The subsequent integers will be x + 2, x + 4, x + 6,... x + 2(n-1).

The sum of these n integers can be represented as:

x + (x + 2) + (x + 4) + ... + (x + 2(n-1)) = S

This can be simplified using the formula for the sum of an arithmetic series:

nx + 2(1 + 2 + ... + (n-1)) = S

The sum of the integers from 1 to (n-1) is given by (n-1)n/2. Substituting this into the equation, we get:

nx + 2[(n-1)n/2] = S nx + n(n-1) = S n(x + n - 1) = S

This equation allows us to solve for x (the first odd integer) given the number of integers (n) and their sum (S). You can then calculate the remaining integers in the sequence.

Alternative Approach: Using Arithmetic Series

Another way to approach this problem is by recognizing that the three consecutive odd integers form an arithmetic series with a common difference of 2. The sum of an arithmetic series can be calculated using the formula:

S = (n/2) * [2a + (n-1)d]

Where:

S = sum of the series (117 in our case)
n = number of terms (3 in our case)
a = first term (what we need to find)
d = common difference (2 in our case)

Plugging in the values, we get:

117 = (3/2) * [2a + (3-1)2] 117 = (3/2) * [2a + 4] 234 = 6a + 12 222 = 6a a = 37

This gives us the first term, 37, and the rest of the sequence follows as before.

Frequently Asked Questions (FAQ)

Q1: Can this method be used for consecutive even integers?

A1: Absolutely! The only difference is that you would represent the integers as x, x + 2, x + 4, etc., and then set up the equation accordingly. The principles of solving the equation remain the same.

Q2: What if I want to find four consecutive odd integers that sum to a specific number?

A2: The approach is similar. You would represent the integers as x, x + 2, x + 4, x + 6, and then set up the equation x + (x + 2) + (x + 4) + (x + 6) = S (where S is the desired sum). Solving this equation would yield the first integer, and you can then determine the rest of the sequence.

Q3: Are there always three consecutive odd integers that sum to any given number?

A3: No. The given number must be divisible by 3, and the result must be an even number. For example, if the sum was 118, you wouldn't find three consecutive odd integers to add up to it.

Q4: Can I use this method for consecutive integers (not just odd or even)?

A4: Yes! For consecutive integers, you would represent them as x, x+1, x+2, etc., and proceed in the same manner.

Q5: What if the sum isn't given, and I need to find it based on the first integer?

A5: If you know the first integer (x), you can easily calculate the sum by adding x, x+2, and x+4 together. The sum will be 3x + 6.

Conclusion

Finding three consecutive odd integers that sum to 117 is a seemingly simple problem, but it offers a valuable opportunity to practice and solidify your understanding of algebraic concepts and problem-solving strategies. By applying algebraic equations, simplifying expressions, and verifying solutions, we have effectively determined that the three consecutive odd integers are 37, 39, and 41. This problem showcases the power of mathematical tools to solve seemingly complex problems through a systematic approach. Understanding these methods empowers you to tackle similar problems with confidence and efficiency, further deepening your mathematical prowess. The generalized solutions provided extend this understanding beyond the specific problem, allowing you to apply the same principles to a wide range of similar scenarios.