Problem 22 - Precalculus Mathematics for Calculus

Question

Prove that \((n+1)^{2}<2 n^{2}\) for all natural numbers \(n \geq 3\)

Step-by-Step Solution

Answer

The inequality holds true for all \(n \geq 3\).

Step 1: Understand the Inequality

We need to prove that \((n+1)^2 < 2n^2\) for all natural numbers \(n \geq 3\). Start by expanding the left side of the inequality.

Step 2: Expand \((n+1)^2\)

Expand the expression: \((n+1)^2 = n^2 + 2n + 1\). Substitute this back into the inequality so it becomes \(n^2 + 2n + 1 < 2n^2\).

Step 3: Rearrange the Inequality

Rearrange the terms: \(n^2 + 2n + 1 < 2n^2\). Subtract \(n^2\) from both sides to get \(2n + 1 < n^2\).

Step 4: Prove The Inequality Holds for \(n \geq 3\)

We need to show that \(n^2 > 2n + 1\) for \(n \geq 3\). Substitute \(n = 3\) into the inequality: \(3^2 = 9\) and \(2(3) + 1 = 7\), hence \(9 > 7\), which holds true. Now, show that \(n^2 > 2n + 1\) becomes more valid as \(n\) increases: - Observe that the function \(n^2\) increases faster than \(2n + 1\) as \(n\) increases because \(\frac{d}{dn}(n^2) = 2n > 2 = \frac{d}{dn}(2n + 1)\) for \(n > 1\).

Step 5: Conclude the Proof

Since \(n^2 > 2n + 1\) holds true for \(n = 3\) and the difference \(n^2 - 2n - 1\) increases as \(n\) gets larger, the original inequality \((n+1)^2 < 2n^2\) holds for all natural numbers \(n \geq 3\). Therefore, we have successfully proven the inequality.

Key Concepts

Understanding Natural Numbers

Natural numbers are the set of positive integers starting from 1 and extending to infinity. These are the numbers we first learn to count with: 1, 2, 3, and so on. They don't include zero or negative numbers.

• Natural numbers are often denoted by the symbol \( \mathbb{N} \).
• In mathematics, operations and properties such as addition and multiplication are first defined using natural numbers.
• Natural numbers are the simplest form of numbers used to count and order.

When we work with natural numbers in proofs, especially with inequalities like this exercise, we focus on their properties like divisibility and order. In this exercise, we are dealing with natural numbers greater than or equal to 3. This allows using specific strategies like mathematical induction to demonstrate properties and inequalities.

The Power of Mathematical Induction

Mathematical induction is a powerful proof technique often used to establish properties of natural numbers. It's like falling dominoes, where showing the first one falls implies all subsequent ones will too.
Here's how it works:

• Base Case: Verify the property is true for the first natural number in the domain, typically \( n = 1 \) or any other starting point.
• Inductive Step: Assume the property holds for some arbitrary natural number \( k \), and then prove it holds for \( k+1 \).

In this exercise, we don't need full induction because we only need to prove the inequality holds for \( n \geq 3 \). However, the concept is important as it gives us confidence that if a property holds for one value and can be shown for the next, it holds for all succeeding natural numbers within that range.

Mastering Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions to solve equations or prove inequalities, just like in this exercise.In our inequality proof, these steps are crucial:

• Expanding Expressions: We expanded \((n+1)^2\) to \(n^2 + 2n + 1\) to simplify the problem.


• Rearranging Terms: After expansion, we subtract \(n^2\) from both sides of the inequality to simplify to \(2n + 1 < n^2\).


• Observing Growth Rates: By comparing \(n^2\) and \(2n + 1\), we analytically see how \(n^2\) grows faster ensuring the inequality holds.

Effective algebraic manipulation begins with small steps like distributing terms and gathering like terms. It's about pushing through the complexity to reveal the simplicity of the solution, making it look almost effortless once solved.