/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Prove that there is a positive i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 10

Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?

Short Answer

Expert verified
n = 1 is a positive integer that equals the sum of the positive integers not exceeding it. The proof is constructive.

Step by step solution

01

- Define the Problem

We need to find a positive integer, say n, such that n is equal to the sum of all positive integers not exceeding it. This means finding n for which \[ n = 1 + 2 + 3 + \cdots + (n-1) + n \]
02

- Use the Sum Formula

Recall the formula for the sum of the first k positive integers: \[ \text{Sum} = \frac{k(k+1)}{2} \]. We need n to be such that \[ n = \frac{n(n+1)}{2}. \]
03

- Set Up the Equation

Setting our equation from step 2, we get: \[ n = \frac{n(n+1)}{2}. \] Multiply both sides by 2 to clear the fraction: \[ 2n = n(n + 1). \]
04

- Simplify the Equation

Rearrange the equation to isolate n: \[ 2n = n^2 + n. \] Then move all terms to one side to form a quadratic equation: \[ n^2 - n = 0. \]
05

- Solve the Quadratic Equation

Factor out n from the equation: \[ n(n - 1) = 0. \] This gives us two solutions: \[ n = 0 \text{ or } n = 1. \] Since we are looking for a positive integer, we select \[ n = 1. \]
06

- Verify the Solution

Check that n = 1 satisfies the initial condition: \[ 1 = 1, \] which is true. Hence, n = 1 is indeed a solution.
07

- Proof Type

Since we provided a specific example that meets the condition, the proof is constructive.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Positive Integers
When we refer to positive integers, we mean numbers that are greater than zero and do not contain any decimal or fractional part. Examples of positive integers include 1, 2, 3, and so on. Understanding positive integers is fundamental when working on many mathematical problems. They are straightforward and matter because they form the basic building blocks of number theory and are used to count discrete objects.
Sum Formula
The sum formula for the first k positive integers is a handy tool in mathematics. It allows you to quickly calculate the sum of any series of consecutive positive integers without having to add each number individually. The formula is: \[ \text{Sum} = \frac{k(k+1)}{2} \]
This formula works by pairing numbers from opposite ends of the range, simplifying the calculation. For example, for the first four positive integers: 1 + 2 + 3 + 4, you can pair them as (1+4) and (2+3), which both sum to 5. Multiplying the number of pairs (2) by 5 gets you 10, which indeed is the sum. Hence, the formula simplifies the addition of large sequences of numbers effortlessly.
Quadratic Equation
A quadratic equation is a polynomial equation of the form \[ ax^2 + bx + c = 0 \]
In our exercise, we derived a quadratic equation while searching for a positive integer, n, where n is equal to the sum of all integers up to and including n. We simplified our equation to: \[ n^2 - n = 0 \]
Solving a quadratic equation often involves factoring, completing the square, or using the quadratic formula. In this case, we factored it as: \[ n(n-1) = 0 \]
This gave solutions n = 0 and n = 1. Since we needed positive integers, we took n = 1 as our solution. Quadratic equations often have two solutions due to the nature of the squared term.
Proof Verification
Proof verification is an essential part of any mathematical proof process. It's what ensures the correctness and reliability of the results obtained. After solving the quadratic equation, we checked if our solution indeed met the initial condition of the problem. In our case, we wanted to ensure that 1 equals the sum of the positive integers not exceeding it.
By substituting n = 1 back into the formula, we verified: \[ 1 = 1 \]
This was true, confirming that our solution was correct. Additionally, we characterized our proof as constructive because we provided a specific example 鈥� n = 1, rather than just showing that at least one such number exists abstractly.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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.

Use resolution to show that the hypotheses 鈥淚t is not raining or Yvette has her umbrella,鈥� 鈥淵vette does not have her umbrella or she does not get wet,鈥� and 鈥淚t is raining or Yvette does not get wet鈥� imply that 鈥淵vette does not get wet.鈥�

Prove that there are no solutions in integers \(x\) and \(y\) to the equation \(2 x^{2}+5 y^{2}=14 .\)

Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), and R(x) be the statements 鈥渪 is a professor,鈥� 鈥渪 is ignorant,鈥� and 鈥渪 is vain,鈥� respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), and R(x), where the domain consists of all people. $$ \begin{array}{l}{\text { a) No professors are ignorant. }} \\ {\text { b) All ignorant people are vain. }} \\ {\text { c) No professors are vain. }} \\\ {\text { d) Does (c) follow from (a) and (b)? }}\end{array} $$

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.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.