Given That $n$ Is An Integer And $n \ \textgreater \ 1$, Prove Algebraically That N 2 − ( N − 2 ) 2 − 2 N^2 - (n-2)^2 - 2 N 2 − ( N − 2 ) 2 − 2 Is Always An Even Number.

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Introduction

In this article, we will delve into the world of algebra and explore a mathematical expression that is always even. The expression in question is $n^2 - (n-2)^2 - 2$, where $n$ is an integer greater than 1. Our goal is to prove algebraically that this expression is always an even number.

Understanding the Expression

Before we dive into the proof, let's break down the expression and understand its components. The expression consists of three main parts:

1. $n^2$: This is the square of the integer $n$.
2. $(n-2)^2$: This is the square of the integer $n-2$.
3. $-2$: This is a constant term that is subtracted from the expression.

Expanding the Expression

To prove that the expression is always even, we need to expand it and simplify it. Let's start by expanding the squared terms:

$n^2 - (n-2)^2 - 2$

$= n^2 - (n^2 - 4n + 4) - 2$

$= n^2 - n^2 + 4n - 4 - 2$

$= 4n - 6$

Simplifying the Expression

Now that we have expanded the expression, let's simplify it further. We can see that the expression is now in the form of $4n - 6$. Since $n$ is an integer, $4n$ is also an integer. Therefore, the expression can be written as:

$4n - 6 = 4(n - 1.5)$

Proving the Expression is Always Even

Now that we have simplified the expression, let's prove that it is always even. We can do this by showing that the expression is a multiple of 2. Since $n$ is an integer, $n - 1.5$ is also an integer. Therefore, $4(n - 1.5)$ is also an integer.

Conclusion

In conclusion, we have proven algebraically that the expression $n^2 - (n-2)^2 - 2$ is always an even number. We expanded the expression, simplified it, and showed that it is a multiple of 2. This proof demonstrates the power of algebra in mathematics and provides a clear understanding of the properties of even numbers.

Properties of Even Numbers

Before we conclude, let's take a moment to discuss the properties of even numbers. An even number is a number that can be divided by 2 without leaving a remainder. In other words, an even number is a number that is a multiple of 2.

Examples of Even Numbers

Some examples of even numbers include:

• 2
• 4
• 6
• 8
• 10

Why is it Important to Understand Even Numbers?

Understanding even numbers is important for a variety of reasons. For example, even numbers are used in many mathematical operations, such as addition and multiplication. Additionally, even numbers are used in many real-world applications, such as finance and engineering.

Real-World Applications of Even Numbers

Even numbers have many real-world applications. For example:

• In finance, even numbers are used to calculate interest rates and investment returns.
• In engineering, even numbers are used to design and build structures, such as bridges and buildings.
• In computer science, even numbers are used to represent binary data and perform calculations.

Conclusion

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Introduction

In our previous article, we proved algebraically that the expression $n^2 - (n-2)^2 - 2$ is always an even number. In this article, we will answer some frequently asked questions about this proof and provide additional insights into the properties of even numbers.

Q: What is the significance of the expression $n^2 - (n-2)^2 - 2$?

A: The expression $n^2 - (n-2)^2 - 2$ is significant because it demonstrates the power of algebra in mathematics. By expanding and simplifying the expression, we can show that it is always an even number, which has important implications for mathematical operations and real-world applications.

Q: Why is it important to understand even numbers?

A: Understanding even numbers is important because they are used in many mathematical operations, such as addition and multiplication. Additionally, even numbers are used in many real-world applications, such as finance and engineering.

Q: Can you provide more examples of even numbers?

A: Yes, some examples of even numbers include:

• 2
• 4
• 6
• 8
• 10
• 12
• 14
• 16

Q: How do you determine if a number is even or odd?

A: To determine if a number is even or odd, you can use the following rule:

• If the number ends in 0, 2, 4, 6, or 8, it is even.
• If the number ends in 1, 3, 5, 7, or 9, it is odd.

Q: Can you provide a real-world example of how even numbers are used in finance?

A: Yes, here is an example:

• Suppose you have a savings account with a balance of $1000. If you deposit $500 into the account, the new balance will be $1500, which is an even number.

Q: Can you provide a real-world example of how even numbers are used in engineering?

A: Yes, here is an example:

• Suppose you are designing a bridge that needs to support a load of 2000 pounds. If you use a beam with a cross-sectional area of 10 square inches, the stress on the beam will be 200 pounds per square inch, which is an even number.

Q: Can you provide a real-world example of how even numbers are used in computer science?

A: Yes, here is an example:

• Suppose you are writing a program that needs to perform a binary operation on two numbers. If the numbers are represented as binary digits (0s and 1s), the result of the operation will always be an even number.

Conclusion

In conclusion, we have answered some frequently asked questions about the proof that $n^2 - (n-2)^2 - 2$ is always an even number. We have also provided additional insights into the properties of even numbers and their importance in mathematical operations and real-world applications.