Proving the Inequality ( n^2 ge 2n 1 ) for ( n ge 3 )

Understanding and proving mathematical inequalities is a critical skill in algebra and higher mathematics. In this article, we will explore how to prove the inequality ( n^2 ge 2n 1 ) for all ( n ge 3 ) using a combination of direct algebraic manipulation and mathematical induction. This inequality, though seemingly simple, can have significant implications in various fields, including computer science and number theory.

Direct Algebraic Manipulation

To begin with, let's start with the left-hand side of the inequality:

[ n^2 ]

We aim to show that this is greater than or equal to the right-hand side:

[ 2n 1 ]

We will manipulate the left-hand side to see if it can be made to match or exceed the right-hand side.

Manipulation and Comparison

Let's consider the expression ( n^2 ) and compare it to ( 2n 1 ).

Step 1: Start with the expression ( n^2 ). We need to show that: Step 2: ( n^2 ge 2n 1 ). To do this, we can rearrange the terms: Step 3: ( n^2 - 2n - 1 ge 0 ).

This is equivalent to showing that ( n^2 - 2n - 1 ) is non-negative for ( n ge 3 ).

Mathematical Induction

Mathematical induction is a powerful tool to prove statements for all natural numbers, particularly useful for inequalities. We will use this method to prove our statement.

Base Case

First, we check the base case ( n 3 ).

[ 3^2 9 ] [ 2(3) 1 7 ]

Clearly, ( 9 ge 7 ), so the base case holds.

Inductive Step

Assume the statement is true for some ( k ge 3 ), i.e., ( k^2 ge 2k 1 ). We need to show that this implies the statement is true for ( k 1 ).

Consider the expression for ( (k 1)^2 ):

[ (k 1)^2 k^2 2k 1 ]

By the inductive hypothesis, ( k^2 ge 2k 1 ). Substituting this into the expression, we get:

[ (k 1)^2 k^2 2k 1 ge (2k 1) 2k 1 ] [ (k 1)^2 ge 4k 2 ]

Now, we need to show that ( 4k 2 ge 2(k 1) 1 ).

[ 4k 2 ge 2k 2 1 ] [ 4k 2 ge 2k 3 ]

Since ( 4k 2 ge 2k 3 ) for all ( k ge 1 ), the statement holds for ( k 1 ).

Conclusion

We have shown that the inequality ( n^2 ge 2n 1 ) holds for ( n 3 ) and that if it holds for ( k ge 3 ), it also holds for ( k 1 ). Therefore, by mathematical induction, the inequality ( n^2 ge 2n 1 ) is true for all ( n ge 3 ).

Applications and Insights

This inequality has various applications. In computer science, it can be used to analyze the efficiency of certain algorithms. For instance, the inequality can help in bounding the number of operations in a computational process. In number theory, it can help in proving properties of integers and sequences.

Understanding and mastering the techniques of algebraic manipulation and mathematical induction are crucial for tackling more complex problems in mathematics and related fields. By practicing such proofs, students can develop a deeper understanding of mathematical concepts and improve their analytical skills.