Problem 20 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Define the Expression

We start by expressing the inequality: \((n+1)^2 < 2n^2\). Expanding the left side gives us: \((n+1)^2 = n^2 + 2n + 1\).

Step 2: Simplify the Inequality

Rewrite the inequality by substituting the expanded form of \((n+1)^2\) into the original inequality: \[ n^2 + 2n + 1 < 2n^2 \]

Step 3: Rearrange the Inequality

Move all terms to one side to form an equation: \[ 0 < 2n^2 - n^2 - 2n - 1 \] Simplifying gives: \[ 0 < n^2 - 2n - 1 \]

Step 4: Analyze the Quadratic Expression

The inequality is now in the form \(n^2 - 2n - 1 > 0\). To solve it, we can find its roots using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -2\), and \(c = -1\).

Step 5: Find the Roots of the Quadratic

Calculate the discriminant: \(b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot (-1) = 4 + 4 = 8\). So, the roots are: \(n = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2}\).

Step 6: Determine Validity for Natural Numbers

Since \(1-\sqrt{2}\) is negative and doesn't apply to natural numbers, we consider \(n > 1 + \sqrt{2}\). Since \(\sqrt{2} \approx 1.41\), \(1 + \sqrt{2} \approx 2.41\), meaning \(n\) must be at least 3 for the inequality \(n^2 - 2n - 1 > 0\) to hold true.

Key Concepts

Natural Numbers

Natural numbers are one of the most basic sets of numbers in mathematics. They refer to positive integers starting from 1, extending infinitely. They are typically the numbers we first learn to count with, such as 1, 2, 3, and so on. In mathematical notation, natural numbers are often represented as \( \mathbb{N} \).

• They do not include zero or negative numbers by traditional definition.
• Natural numbers are closed under addition and multiplication, meaning adding or multiplying any two natural numbers returns a natural number.
• They form the basis for more complex mathematical concepts and applications.

In the problem of proving the inequality \((n+1)^{2} < 2 n^{2}\) for \(n \geq 3\), the natural numbers applicable are 3 and above. This ensures that 1 and negative results of computations are not considered valid solutions.

Quadratic Formula

The quadratic formula is a crucial mathematical tool used to find the roots of a quadratic equation, which are equations of the form \(ax^2 + bx + c = 0\).

• The formula is represented as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• It derives from the process of completing the square, offering a universal method to solve any quadratic equation.
• The discriminant, \(b^2 - 4ac\), determines the nature of the roots: if the discriminant is positive, there are two distinct real roots; if zero, exactly one real root; and if negative, there are no real roots.

In our specific problem, we employed the quadratic formula to assess the inequality \(n^2 - 2n - 1 > 0\). This helps find out for which values of \(n\) the expression is greater than zero.

Quadratic Roots

Quadratic roots are the solutions of a quadratic equation, obtained when the equation equals zero. They are specifically the values of \(x\) where \(ax^2 + bx + c = 0\) holds true.

• The roots may be real or complex, depending on the value of the discriminant \(b^2 - 4ac\).
• For real roots, these can be equal or distinct. In the situation of distinct roots, they occupy two points on the x-axis.
• The process of finding roots typically involves factoring, completing the square, or using the quadratic formula.

For this inequality proof, the roots \(n=1 \pm \sqrt{2}\) were calculated using the quadratic formula. We used these to determine the range of natural numbers (3 and above) where the inequality \(n^2 - 2n - 1 > 0\) holds true.

Expanding Expressions

Expanding expressions is a technique used to simplify or rewrite algebraic terms by distributing multiplication over addition or subtraction.

• It involves applying the distributive property \(a(b + c) = ab + ac\).
• Expansion helps to make complex algebraic expressions more manageable and can reveal patterns and simplifications that are not immediately obvious.
• It is essential in transforming polynomial equations such as \( (n+1)^2 = n^2 + 2n + 1 \).

In the context of this problem, expanding \((n+1)^2\) allowed us to manipulate and compare it effectively with \(2n^2\), ultimately leading us to an inequality of a simpler form \(n^2 - 2n - 1 > 0\). This simplification was a key step in evaluating which values of \(n\) satisfy the given inequality.