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Chapter 1: Problem 1

Prove that \(n^{2}+1 \geq 2^{n}\) when \(n\) is a positive integer with \(1 \leq n \leq 4 .\)

Short Answer

Expert verified
The inequality holds for all positive integers from 1 to 4.

Step by step solution

01

Check for n = 1

Substitute n = 1 into the inequality. Verify that the inequality holds.\(n^2 + 1\geq 2^{n}\)For n = 1:\(1^2 + 1 = 1 + 1 = 2\)\(2^{1} = 2\)Thus, \(1^2 + 1 = 2\) and \(2^{1} = 2\), so \(2 \geq 2\) holds true.
02

Check for n = 2

Substitute n = 2 into the inequality. Verify that the inequality holds.\(n^2 + 1\geq 2^{n}\)For n = 2:\(2^2 + 1 = 4 + 1 = 5\)\(2^{2} = 4\)Thus, \(2^2 + 1 = 5\) and \(2^{2} = 4\), so \(5 \geq 4\) holds true.
03

Check for n = 3

Substitute n = 3 into the inequality. Verify that the inequality holds.\(n^2 + 1\geq 2^{n}\)For n = 3:\(3^2 + 1 = 9 + 1 = 10\)\(2^{3} = 8\)Thus, \(3^2 + 1 = 10\) and \(2^{3} = 8\), so \(10 \geq 8\) holds true.
04

Check for n = 4

Substitute n = 4 into the inequality. Verify that the inequality holds.\(n^2 + 1\geq 2^{n}\)For n = 4:\(4^2 + 1 = 16 + 1 = 17\)\(2^{4} = 16\)Thus, \(4^2 + 1 = 17\) and \(2^{4} = 16\), so \(17 \geq 16\) holds true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

positive integers
A positive integer is any whole number greater than zero. They are the numbers you typically count with and include numbers like 1, 2, 3, and so on.

Understanding the concept of positive integers is crucial in many areas of mathematics. They form the backbone of number theory and are often the first numbers we learn to work with.

In this exercise, all our values for \(n\) are positive integers. Specifically, we are asked to consider \(n\) values between 1 and 4. This is important because positive integers have unique properties that can affect how inequalities and other mathematical rules apply to them. For instance, since there are no fractions or decimals, each calculation involves straightforward whole numbers.
mathematical induction
Mathematical induction is a powerful proof technique used to establish the truth of an infinite number of statements. It's often compared to dominoes falling because proving one statement helps you prove the next, and so on.

The process of mathematical induction involves two main steps:
  • **Base Case**: Verify the statement is true for the initial value (often \(n = 1\)).
  • **Inductive Step**: Show that if the statement holds for some positive integer \(k\), then it also holds for \(k+1\).

This technique ensures that if the base case is true and the inductive step works, the statement is true for all positive integers starting from the base case.

In our exercise, proving the base cases for \(n = 1\) through \(n = 4\) uses a similar approach but without the full induction step. However, understanding induction helps you get why checking individual cases is sometimes necessary.
base case checking
Base case checking is the first step in mathematical induction. It establishes that the statement holds for the smallest value in the domain, often \(n = 1\). This step is crucial because it serves as the foundation for proving the statement for all subsequent values.

Here's a breakdown of the steps in our exercise:
  • For \(n = 1\), we substitute 1 into the inequality \(n^2 + 1 \geq 2^n\) and verify it holds since 2 equals 2.
  • For \(n = 2\), substituting 2 results in 5 being greater than 4, showing the inequality is true.
  • For \(n = 3\), we find 10 is greater than 8, again supporting the inequality.
  • For \(n = 4\), substituting 4 yields 17 greater than 16, proving the base case for all four values.

By confirming these cases, we build confidence that the inequality holds true, laying the groundwork for more comprehensive proofs. Base case checking is an essential part of ensuring logical consistency in mathematical arguments.

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Most popular questions from this chapter

Express the negations of these propositions using quantifiers, and in English. a) Every student in this class likes mathematics. b) There is a student in this class who has never seen a computer. c) There is a student in this class who has taken every mathematics course offered at this school. d) There is a student in this class who has been in at least one room of every building on campus.

Prove that there are no solutions in positive integers \(x\) and \(y\) to the equation \(x^{4}+y^{4}=625\)

For each of these arguments, explain which rules of inference are used for each step. a) 鈥淟inda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket.鈥� b) 鈥淓ach of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year.鈥� c) 鈥淎ll movies produced by John Sayles are wonder-ful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners.鈥� d) 鈥淭here is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre.鈥�

Express each of these statements using predicates and quantifiers. a) A passenger on an airline qualifies as an elite flyer if the passenger flies more than 25,000 miles in a year or takes more than 25 flights during that year. b) A man qualifies for the marathon if his best previous time is less than 3 hours and a woman qualifies for the marathon if her best previous time is less than 3.5 hours. c) A student must take at least 60 course hours, or at least 45 course hours and write a master鈥檚 thesis, and receive a grade no lower than a B in all required courses, to receive a master鈥檚 degree. d) There is a student who has taken more than 21 credit hours in a semester and received all A鈥檚.

Prove that there is no positive integer \(n\) such that \(n^{2}+\) \(n^{3}=100 .\)
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