Problem 22 - 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 \(n \geq 3\) as \(n^{2} - 2n - 1 > 0\) for these values.

Step 1: Understand the Inequality

Begin with the inequality \((n+1)^{2}<2n^{2}\). This inequality suggests that the square of \(n+1\) is less than twice the square of \(n\). Our task is to prove that this holds for all natural numbers \(n\) greater than or equal to 3.

Step 2: Expand the Left Side

Expand the left side of the inequality: \((n+1)^{2} = n^{2} + 2n + 1\). Therefore, the original inequality becomes \(n^{2} + 2n + 1 < 2n^{2}\).

Step 3: Simplify the Inequality

Subtract \(n^{2}\) from both sides to get the inequality in a simpler form: \(2n + 1 < n^{2}\).

Step 4: Rearrange to Compare Terms

Rearrange the inequality to compare the terms: \(n^{2} - 2n - 1 > 0\). This new inequality needs to be true for all \(n \geq 3\).

Step 5: Analyze and Prove the Inequality Holds

Let's evaluate the expression \(n^{2} - 2n - 1\) for \(n=3\) to check if it holds: \(n^{2} - 2n - 1 = 3^2 - 2(3) - 1 = 9 - 6 - 1 = 2 > 0\). Thus, it holds for \(n=3\). To prove this expression remains positive for larger \(n\), note that \(n^{2}\) grows faster than \(2n + 1\) as \(n\) increases because the quadratic term \(n^{2}\) will dominate for large \(n\).

Step 6: Conclude the Mathematical Induction for Large n

For \(n \geq 3\), since we demonstrated that the inequality holds for \(n=3\) and observed that the expression \(n^{2} - 2n - 1\) grows as \(n\) increases, it's reasonable to conclude that the inequality \((n+1)^{2}<2n^{2}\) holds for all \(n \geq 3\) by mathematical induction.

Key Concepts

Natural Numbers

Natural numbers are the set of positive integers starting from 1 and continuing indefinitely: 1, 2, 3, and so on. They are the most basic numbers used in everyday math. Natural numbers allow us to count objects, represent quantities, and perform basic arithmetic operations like addition and multiplication.

In many mathematical problems, we begin examining patterns or proving statements using the natural numbers. In the context of this exercise, we're interested in proving an inequality starting from the natural number 3 and onward. What's fascinating about natural numbers is how they lay the foundation for more complex structures such as integers and real numbers. They also provide a simple, straightforward way to model many real-world situations.

Understanding the properties of natural numbers, such as their order and how they increase by one each step, is essential for tackling any math problem that involves counting or sequences.

Expanding Binomials

Expanding binomials involves taking an expression in the form of ext{(a + b)} raised to a power and rewriting it in an expanded form. This process helps us simplify expressions and is a useful tool in algebra for solving inequalities and equations.

For the inequality we're exploring, the expansion of ext{(n+1)}^2 is necessary. By expanding it, we convert ext{(n+1)}^2 into n^2 + 2n + 1 . This makes it easier to compare to other terms and understand the relationship in the inequality. Thanks to expansion, we can bring ext{polynomial} difficulties to a more straightforward level of elementary algebra. This technique is not only used in simple squaring but also in higher powers, following conventions like the binomial theorem. Remember, carefully expanding and simplifying expressions opens the door to understanding more complex relationships within algebraic contexts.

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is related to a value with either less than, greater than, or equal signs. These can be expressed in the standard form as ax^2 + bx + c > 0 , ax^2 + bx + c < 0 , or with other comparison symbols. Solving such inequalities is crucial for analyzing ranges of values and understanding the behavior of functions over certain domains.

In our example, after simplifying and rearranging the expanded binomial inequality, we end up with a quadratic inequality: n^2 - 2n - 1 > 0 . Solving quadratic inequalities often requires factoring or applying methods like completing the square or using the quadratic formula. The goal is to find where the inequality is satisfied, usually considering where the expression crosses zero, which helps us understand where it is positive or negative.

This particular inequality shows that as n increases, n^2 grows faster than 2n + 1 , causing the inequality to hold true for all natural numbers greater than or equal to 3.

Mathematical Induction

Mathematical induction is a powerful proof technique used to verify statements about natural numbers. It follows a specific logical structure that involves two main steps: the base case and the induction step.

First, in the **base case**, we establish that the statement is true for the initial value of the sequence, often n = 1 or another starting point such as n = 3 in our exercise. This provides the foundation for our proof.

Next, in the **induction step**, we assume the statement holds for some arbitrary natural number k and prove that if it holds for k , it must also hold for k+1 . This creates a "domino effect" proving the statement for all natural numbers beyond the base case.

In this exercise, though not explicitly applied step-by-step, we use the **principle of induction** by demonstrating that the inequality holds for n = 3 , and recognizing how the quadratic nature guarantees it continues to hold as n increases, covering all natural numbers starting from 3 onward. Induction is invaluable in proving properties and inequalities across an infinite set like the natural numbers.