Introduction to the Problem
The statement "Prove ( N^2 \geq N ) for all integers ( N \geq 2 )" is an assertion concerning the properties of positive integers. Understanding and proving mathematical inequalities is a fundamental aspect of number theory, and this particular case allows for an exploration of basic algebra and the behavior of quadratic expressions.
Analyzing the Inequality
First, let’s simplify the inequality we want to prove:
We start by rewriting the given inequality:
[N^2 \geq N
]
Subtracting ( N ) from both sides gives us:
[N^2 – N \geq 0
]
This expression can be factored as follows:
[N(N – 1) \geq 0
]
Evaluating the Factors
To evaluate the inequality ( N(N – 1) \geq 0 ), it is important to consider the values of ( N ).
-
Identify the Critical Points: The factors of the expression lead to two critical points, ( N = 0 ) and ( N = 1 ). These points provide boundaries where the expression may change signs.
- Determine the Sign of the Product:
- When ( N < 0 ): Both ( N ) and ( (N – 1) ) are negative, resulting in a positive product, but this range is outside our interest since ( N ) must be at least 2.
- When ( N = 0 ): The product equals zero.
- When ( N = 1 ): The product equals zero.
- When ( N > 1 ): Here, ( N ) is positive and ( (N – 1) ) is also positive, which makes the product ( N(N – 1) ) positive.
Since we focus specifically on integers ( N \geq 2 ), we notice that both factors will be positive for these values.
Practical Verification
To reinforce the proof, we can check specific values of ( N ):
-
For ( N = 2 ):
[
N^2 = 4 \quad \text{and} \quad N = 2 \quad \Rightarrow \quad 4 \geq 2 \quad \text{(True)}
] -
For ( N = 3 ):
[
N^2 = 9 \quad \text{and} \quad N = 3 \quad \Rightarrow \quad 9 \geq 3 \quad \text{(True)}
] - For ( N = 4 ):
[
N^2 = 16 \quad \text{and} \quad N = 4 \quad \Rightarrow \quad 16 \geq 4 \quad \text{(True)}
]
This testing continues to hold true for larger integers, supporting the inequality for all integers ( N \geq 2 ).
Summary of the Proof
The inequality ( N^2 \geq N ) proves true for all integers greater than or equal to 2. From factoring ( N(N-1) ), the critical points illustrate that for ( N ) values of 2 and above, both factors are non-negative, confirming that the product is non-negative. Therefore, the assertion that ( N^2 \geq N ) stands validated.
FAQ Section
1. What is the significance of the inequality ( N^2 \geq N ) in mathematics?
This inequality illustrates a basic property of quadratic functions and serves as a foundation for understanding more complex mathematical concepts, including polynomial behavior and roots.
2. Are there any integers for which ( N^2 < N ) when ( N \geq 2 )?
No, for all integers starting from 2 and upwards, ( N^2 ) is always greater than or equal to ( N ). Values below 2 do not apply within the scope of this inequality.
3. Can this proof method be used for other polynomial inequalities?
Yes, the approach of factoring and analyzing sign changes is a common technique in mathematics for proving polynomial inequalities, and it can be applied to various forms beyond this basic case.