Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-32: Find the smallest positive integer \(j\) for which the statement is true. Use the extended principle of mathematical induction to prove that the formula is true for every integer greater than \(j\). $$ n^{2}+18 \leq n^{3} $$

Step-by-Step Solution

Answer

The smallest positive integer \( j \) is 3.

Step 1: Understand the Inequality

We need to find the smallest positive integer \( j \) such that the inequality \( n^{2} + 18 \leq n^{3} \) holds true for every integer \( n \geq j \).

Step 2: Test Initial Values

Begin by testing small integer values of \( n \).\1. For \( n = 1 \): \( 1^2 + 18 = 19 \) and \( 1^3 = 1 \). Clearly, \( 19 \) is not \( \leq 1 \).\2. For \( n = 2 \): \( 2^2 + 18 = 22 \) and \( 2^3 = 8 \). Clearly, \( 22 \) is not \( \leq 8 \).\3. For \( n = 3 \): \( 3^2 + 18 = 27 \) and \( 3^3 = 27 \). Clearly, \( 27 \leq 27 \).\The inequality holds true starting from \( n = 3 \).

Step 3: Set the Base Case

Let \( j = 3 \). We know the base case \( n = 3 \) satisfies \( n^{2} + 18 \leq n^{3} \). So, the base case of our induction is satisfied.

Step 4: Perform the Induction Step

Assume that the statement holds for some integer \( k \geq 3 \), that is, \( k^{2} + 18 \leq k^{3} \). We need to show that it also holds for \( k + 1 \).\\( (k+1)^2 + 18 \leq (k+1)^3 \).

Step 5: Simplifying the Induction Step

Expand and simplify the inequality for \( k+1 \):\- \( (k+1)^2 + 18 = k^2 + 2k + 1 + 18 \).\- \( (k+1)^3 = k^3 + 3k^2 + 3k + 1 \).\We assume \( k^2 + 18 \leq k^3 \) because of our induction hypothesis.

Step 6: Compare Terms

Subtract \( k^2 \) from both sides: \\( 2k + 19 \leq 3k^2 + 3k + 1 - k^3 \). \Simplify this inequality further: \\( 19 \leq (3k^2 - k^3) + k + 1 \).\Given \( k \geq 3 \), the dominant term is \( k^3 \), ensuring this inequality holds true.

Step 7: Conclusion

Since we have shown the base case is true and our induction step holds, the inequality \( n^{2}+18 \leq n^{3} \) is true for all integers \( n \geq j \) where \( j = 3 \). Therefore, the smallest positive integer \( j \) is 3.

Key Concepts

Inequality

An inequality in mathematics is a statement that describes the relative size or order of two values. It tells us how one value compares to another.
An inequality is often expressed using symbols such as:

• \( \leq \) which means "less than or equal to"
• \( \geq \) which means "greater than or equal to"

In the given exercise, the inequality \( n^2 + 18 \leq n^3 \) essentially compares the square of a number, plus a fixed constant, with the cube of that number.
This inequality asks us to find out when the expression on the left side becomes smaller or equal to the one on the right side as the number \( n \) increases.
Testing smaller values manually is a common first step. It helps to determine a starting point where the inequality holds true consistently.

Positive Integer

A positive integer is any whole number greater than zero.
The set of positive integers includes numbers like 1, 2, 3, and so on. They are basic building blocks in arithmetic and number theory.
In many mathematical proofs, especially involving induction, we often look at positive integers because they are easily countable and follow a natural order. In the context of mathematical induction, we are usually asked to show something is true for all positive integers greater than a specific value.
In this exercise, we find the smallest positive integer \( j \) for which \( n^2 + 18 \leq n^3 \) holds when \( n \) is a positive integer, meaning \( n \geq j \).
This smallest value, in the given solution, turns out to be 3.

Mathematics Education

Mathematics education involves teaching and learning mathematics concepts in an understandable and accessible way.
It often focuses on developing problem-solving abilities and logical reasoning skills.
Understanding mathematical terms and being able to decipher complex equations like \( n^2 + 18 \leq n^3 \) is crucial.
Using exercises such as the one presented, students develop a deeper understanding of mathematical concepts and learn how to apply them effectively in proofs and calculations.
Mathematics education aims not only to teach computational skills but also to develop a conceptual understanding that allows students to tackle a wide array of mathematical problems.

Proof Technique

A proof technique in mathematics is a systematic method used to establish the validity of a statement or theorem.
One of the powerful proof techniques is mathematical induction, often used to prove statements involving natural numbers. It usually follows these simple stages:

• Base Case: Show the statement is true for the initial positive integer, often 1 or some specified other base value.
• Inductive Step: Assume the statement is true for some integer \( k \), and then demonstrate it holds true for \( k+1 \).

In this exercise, we are using induction to show that the inequality \( n^2 + 18 \leq n^3 \) holds true for every integer \( n \) starting from 3.
Induction works like dominoes: if you can make sure the first domino falls and show that whenever one domino falls, the next does too, then all dominoes will fall.
This powerful technique helps to establish the truth of countless mathematical theorems and is a fundamental part of mathematics education.