Proving n^3>n^2: Using Mathematical Induction and Set of Natural Numbers

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In summary, the conversation discusses how to prove the statement n^3>n^2 for n≥2 using mathematical induction. The speakers suggest using the fact that αn^2>n^2 for any α>1 and noting that n^3-n^2=n^2(n-1), which is always a positive integer as long as n>1. They also mention that this may seem like a short proof, but it is valid.
  • #1
Ed Quanta
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How do you go about proving a statement like n^3>n^2 for n is equal to or greater than 2? I can prove this using mathematical induction, but I am unsure how to show n^3-n^2 is an element of the set of natural numbers without just saying in general that this must always yield a number equal to or greater than 1.
 
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  • #2
Can't you just say that because

[tex]\forall\alpha>1:\alpha n^2 > n^2[/tex]

we must conclude that

[tex]\forall n>1: n^3 > n^2[/tex]
 
  • #3
Why not just note that n3- n2= n2(n-1)? As long as n> 1, this will be a positive integer.
 
  • #4
Yeah, Halls of Ivy, that's what I was thinking. It just looked like too short a proof so I thought I was missing something. Thanks dudes.
 

Related to Proving n^3>n^2: Using Mathematical Induction and Set of Natural Numbers

1. What is an order relation?

An order relation is a mathematical concept that describes the relationship between two elements in a set. It indicates whether one element is greater than, less than, or equal to another element.

2. What are the different types of order relations?

The three main types of order relations are total order, partial order, and strict order. In a total order, all elements in a set can be compared to each other. In a partial order, only some elements can be compared. In a strict order, elements cannot be equal to each other.

3. How is an order relation represented?

An order relation is typically represented using symbols, such as > (greater than), < (less than), and = (equal to). It can also be represented visually using a number line or a Hasse diagram.

4. What is the difference between a total order and a partial order?

In a total order, all elements can be compared to each other, whereas in a partial order, only some elements can be compared. Additionally, in a total order, all elements must be comparable, but in a partial order, some elements may not have a defined relationship with each other.

5. Why is the concept of order relation important?

The concept of order relation is important in mathematics because it allows us to compare elements in a set and establish a sense of order. This is essential in many mathematical concepts, such as sorting, ranking, and understanding patterns and relationships between numbers or objects.

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