Proof (2n + 7)(4n - 3) - 8 Is Always An Odd Number
Introduction
In this article, we delve into a fascinating mathematical problem that revolves around the nature of odd and even numbers. Specifically, we aim to prove that the expression (2n + 7)(4n - 3) - 8 will always yield an odd number for any integer value of n. This exploration not only provides a deeper understanding of number theory but also showcases the elegance and precision of mathematical proofs. We will meticulously break down the expression, simplify it, and demonstrate how it can be represented in the form 2k + 1, where k is an integer, thus unequivocally establishing its odd nature.
Understanding Odd and Even Numbers
Before we dive into the proof, it is crucial to have a firm grasp of the fundamental concepts of odd and even numbers. An even number is any integer that is exactly divisible by 2, meaning it can be expressed in the form 2k, where k is an integer. Examples of even numbers include -4, 0, 2, 6, and 10. Conversely, an odd number is an integer that leaves a remainder of 1 when divided by 2. This means it can be expressed in the form 2k + 1, where k is an integer. Examples of odd numbers include -3, 1, 5, 9, and 13. The key difference lies in the presence of the '+ 1' in the expression for odd numbers, which signifies the remainder that makes them distinct from even numbers.
Understanding this distinction is pivotal for our proof. We aim to manipulate the given expression (2n + 7)(4n - 3) - 8 into the form 2k + 1. This transformation will serve as a definitive demonstration that the expression invariably results in an odd number, regardless of the integer value of n. Our journey involves algebraic simplification, factoring, and a keen eye for recognizing patterns that lead us to the desired form. Let's embark on this mathematical exploration with clarity and precision.
Expanding and Simplifying the Expression
Our journey begins with the expression (2n + 7)(4n - 3) - 8. To prove that this expression always results in an odd number, we must first expand and simplify it. This process will allow us to better understand its structure and identify opportunities to rewrite it in the form 2k + 1. Let's break down the steps involved in this crucial simplification.
Expanding the Product
The first step is to expand the product of the two binomials, (2n + 7) and (4n - 3). We use the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last) to multiply each term in the first binomial by each term in the second binomial:
(2n + 7)(4n - 3) = (2n)(4n) + (2n)(-3) + (7)(4n) + (7)(-3)
Now, let's perform these multiplications:
- (2n)(4n) = 8n²
- (2n)(-3) = -6n
- (7)(4n) = 28n
- (7)(-3) = -21
So, the expanded form of the product is:
8n² - 6n + 28n - 21
Combining Like Terms
Next, we simplify the expanded expression by combining like terms. In this case, we have two terms that involve n: -6n and 28n. Adding these together gives us:
-6n + 28n = 22n
Now, we can rewrite the expression as:
8n² + 22n - 21
Subtracting 8
Finally, we need to subtract 8 from the simplified expression, as per the original problem statement:
(8n² + 22n - 21) - 8 = 8n² + 22n - 29
Therefore, the simplified form of the original expression (2n + 7)(4n - 3) - 8 is 8n² + 22n - 29. This simplified form is much easier to work with and allows us to proceed with our goal of expressing it in the form 2k + 1.
Expressing the Simplified Form as 2k + 1
Having simplified the expression (2n + 7)(4n - 3) - 8 to 8n² + 22n - 29, our next crucial step is to demonstrate that this expression can be written in the form 2k + 1, where k is an integer. This representation will definitively prove that the expression always yields an odd number for any integer value of n. Let's delve into the process of rewriting the expression.
Factoring out 2
The key to expressing an integer in the form 2k + 1 is to isolate a factor of 2. We begin by examining the terms in our simplified expression, 8n² + 22n - 29. We notice that 8n² and 22n are both even, meaning they are divisible by 2. We can factor out a 2 from these terms:
8n² + 22n = 2(4n² + 11n)
This leaves us with:
2(4n² + 11n) - 29
Rewriting -29
Now, we focus on the constant term, -29. To express this in the form required, we need to separate it into an even number and 1. We can rewrite -29 as:
-29 = -30 + 1
Here, -30 is an even number (divisible by 2), and 1 is the remainder we need to show the odd nature of the expression.
Substituting and Rearranging
Substituting -30 + 1 for -29 in our expression, we get:
2(4n² + 11n) - 30 + 1
Now, we can factor out a 2 from -30:
-30 = 2(-15)
Our expression now looks like this:
2(4n² + 11n) + 2(-15) + 1
Finally, we factor out the 2 from the first two terms:
2(4n² + 11n - 15) + 1
Identifying k
Now we have successfully expressed our original simplified expression in the form 2k + 1. In this case, k represents the expression (4n² + 11n - 15). Since n is an integer, 4n², 11n, and -15 are all integers. Therefore, their sum, (4n² + 11n - 15), is also an integer. This confirms that k is an integer.
Conclusion: (2n + 7)(4n - 3) - 8 is Always Odd
In conclusion, we have successfully demonstrated that the expression (2n + 7)(4n - 3) - 8 always results in an odd number for any integer value of n. Our journey began with expanding and simplifying the given expression to 8n² + 22n - 29. This simplification paved the way for the pivotal step of expressing the result in the form 2k + 1, which is the defining characteristic of an odd number.
Proof Summary
We meticulously factored out a 2 from the terms 8n² and 22n, and then rewrote the constant term -29 as -30 + 1. This strategic manipulation allowed us to rewrite the entire expression as 2(4n² + 11n - 15) + 1. By identifying (4n² + 11n - 15) as k, we definitively showed that the expression could be represented as 2k + 1, where k is an integer.
Significance of the Proof
This proof is not just a mathematical exercise; it highlights the elegance and power of mathematical reasoning. By following a logical sequence of steps, we transformed a complex-looking expression into a clear and concise representation that unveiled its inherent nature. The ability to express a number in the form 2k + 1 is a fundamental tool in number theory, allowing us to classify numbers and understand their properties.
Final Thoughts
The result of this proof underscores the consistent and predictable behavior of mathematical expressions. Regardless of the integer value of n, the expression (2n + 7)(4n - 3) - 8 will always yield an odd number. This certainty is a testament to the beauty and rigor of mathematics, where logical deductions lead to unwavering conclusions. The ability to prove such statements is a cornerstone of mathematical understanding and a valuable skill in problem-solving.
Through this exploration, we have not only solved a specific mathematical problem but also reinforced our understanding of odd and even numbers, algebraic manipulation, and the power of mathematical proofs. This journey serves as a reminder of the profound insights that mathematics offers and the satisfaction of arriving at a definitive answer through logical reasoning.
