/*! 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 14 Prove that there do not exist th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 14

Prove that there do not exist three consecutive natural numbers such that the cube of the largest is equal to the sum of the cubes of the other two.

Short Answer

Expert verified
We have to prove that there do not exist three consecutive natural numbers (n, n+1, n+2) such that \((n+2)^3 = n^3 + (n+1)^3\). After setting up and simplifying the equation, we get \(n^3 - 3n^2 + 3n^2 + 3n - 12n = 0\). We can see that the equation cannot be simplified or factored further. Since the left side of the equation is always increasing while the right side remains 0, we can conclude that there do not exist three consecutive natural numbers (n, n+1, n+2) such that the cube of the largest (n+2)^3 is equal to the sum of the cubes of the other two (n^3 + (n+1)^3).

Step by step solution

01

Set up the equation

Write down the equation based on the given condition; that the cube of the largest number is equal to the sum of the cubes of the other two: \((n+2)^3 = n^3 + (n+1)^3\)
02

Expand the cubes

Expand the equation using the binomial theorem to simplify the equation: \((n^3 + 3n^2\cdot 2 + 3n\cdot 2^2 + 2^3) = n^3 + (n^3 + 3n^2 + 3n + 1)\)
03

Combine like terms

Combine like terms on both sides of the equation, including n^3 values and square terms: \(n^3 + 6n^2 + 12n + 8 = n^3 + n^3 + 3n^2 + 3n + 1\)
04

Eliminate duplicate terms and simplify

Eliminate duplicate terms on both sides of the equation and simplify remaining terms to get a clearer view of the relationship between n, n+1, and n+2: \(6n^2 + 12n + 8 = n^3 + 3n^2 + 3n + 1\)
05

Rearrange the equation and try to factor

Rearrange the equation so that all terms are on one side and try to factor any potential common factors: \(n^3 - 3n^2 + 3n^2 + 3n - 12n - 8 + 8 = n^3 - 3n^2 + 3n^2 + 3n - 12n = 0\) After attempting to factor the equation, we can see that there are no common factors that can be simplified.
06

Show that the equation cannot hold true

Since the equation cannot be simplified or factored further, and the left side of the equation is always increasing while the right side remains 0, we can conclude that there do not exist three consecutive natural numbers (n, n+1, n+2) such that the cube of the largest (n+2)^3 is equal to the sum of the cubes of the other two (n^3 + (n+1)^3).

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.

Consecutive Natural Numbers
Natural numbers are the set of positive integers starting from 1. Consecutive natural numbers are numbers that follow each other in sequence without any gaps. For example, 3, 4, and 5 are consecutive natural numbers. They play a crucial role in mathematical patterns and proofs due to their predictable sequence and regularity.

In many mathematical problems, like the one where we need to prove a certain condition involving three consecutive numbers, understanding what consecutive natural numbers are is essential. The numbers can be represented as \( n \), \( n+1 \), and \( n+2 \). Here, \( n \) is any natural number, and the next two terms in the sequence are simply incremented by 1 and 2, respectively.
If these numbers were used in the problem to construct an equation involving powers, such as cubes, it becomes manageable to both express and simplify the equation accurately. Representing the numbers as variables helps in generalizing the problem instead of restricting it to specific values.
Cubes and Cube Roots
When a number is raised to the third power, we refer to it as its cube. For instance, if we take any number, say \( n \), its cube is represented as \( n^3 \). This operation is significantly used in scenarios involving volume, such as finding the volume of a cube where all sides are equal. Cube roots, on the other hand, represent the inverse operation, where you find a number which, when cubed, gives you the original value. For instance, the cube root of 8 is 2, because \( 2^3 = 8 \).

In the context of the problem, the cubes of consecutive natural numbers were considered, specifically \((n+2)^3 \), \(n^3 \), and \((n+1)^3 \). Expanding these expressions using the binomial theorem allowed us to attempt solving the problem by analyzing the relationships between these cubes.
This exercise involved trying to find out if the cube of the largest number \((n+2)^3 \) equals the sum of the cubes of the other two \(n^3+(n+1)^3 \). Though it may initially seem possible, the algebraic manipulation of these cubes leads to a different conclusion.
Proof Techniques
Proof techniques are the methods used by mathematicians to establish the validity of mathematical statements. They are essential in confirming the correctness of ideas and solutions. In this exercise, the aim is to prove that a certain equation cannot hold true for any consecutive natural numbers. Different proof techniques are used depending on the problem structure and context.

  • **Direct proof:** This involves a straightforward explanation based on direct algebraic solutions. It started by forming an equation according to the conditions given.
  • **Indirect proof or proof by contradiction:** Although not directly used here, this method involves assuming the opposite of what you want to prove and demonstrating a contradiction. In our context, the problem implied the assumption that three consecutive numbers could satisfy the condition.
  • **Algebraic manipulation:** Simplifying complex expressions to reach a logical conclusion. By expanding and simplifying, the equation in the solution demonstrated that no values of \(n\) satisfy the condition of the cubes.
Using these proof techniques, it's shown that the equation derived does not work for any natural number \(n\), thereby reinforcing the correctness of the initial assumption about the impossibility.

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

The purpose of this exploration is to investigate the possibilities for which integers cannot be the sum of the cubes of two or three integers. (a) If \(x\) is an integer, what are the possible values (between 0 and 8 , inclusive) for \(x^{3}\) modulo \(9 ?\) (b) If \(x\) and \(y\) are integers, what are the possible values for \(x^{3}+y^{3}\) (between 0 and 8 , inclusive) modulo \(9 ?\) (c) If \(k\) is an integer and \(k \equiv 3(\bmod 9), \operatorname{can} k\) be equal to the sum of the cubes of two integers? Explain. (d) If \(k\) is an integer and \(k \equiv 4(\bmod 9), \operatorname{can} k\) be equal to the sum of the cubes of two integers? Explain. (e) State and prove a theorem of the following form: For each integer \(k\), if (conditions on \(k\) ), then \(k\) cannot be written as the sum of the cubes of two integers. Be as complete with the conditions on \(k\) as possible based on the explorations in Part (b). (f) If \(x, y,\) and \(z\) are integers, what are the possible values (between 0 and 8 , inclusive) for \(x^{3}+y^{3}+z^{3}\) modulo \(9 ?\) (g) If \(k\) is an integer and \(k \equiv 4(\bmod 9),\) can \(k\) be equal to the sum of the cubes of three integers? Explain. (h) State and prove a theorem of the following form: For each integer \(k\), if (conditions on \(k\) ), then \(k\) cannot be written as the sum of the cubes of three integers. Be as complete with the conditions on \(k\) as possible based on the explorations in Part (f).

Determine if each of the following statements is true or false. Provide a counterexample for statements that are false and provide a complete proof for those that are true. (a) For all real numbers \(x\) and \(y, \sqrt{x y} \leq \frac{x+y}{2}\). (b) For all real numbers \(x\) and \(y, x y \leq\left(\frac{x+y}{2}\right)^{2}\). (c) For all nonnegative real numbers \(x\) and \(y, \sqrt{x y} \leq \frac{x+y}{2}\).

Let \(n\) be a natural number. Prove each of the following: (a) For every integer \(a, a \equiv a(\bmod n)\). This is called the reflexive property of congruence modulo \(n\). (b) For all integers \(a\) and \(b,\) if \(a \equiv b(\bmod n),\) then \(b \equiv a(\bmod n)\). This is called the symmetric property of congruence modulo \(n\). (c) For all integers \(a, b,\) and \(c,\) if \(a \equiv b(\bmod n)\) and \(b \equiv c(\bmod n),\) then \(a \equiv c(\bmod n)\) This is called the transitive property of congruence modulo \(n\).

(Exercise (15), Section 3.1) Let \(r\) be a positive real number. The equation for a circle of radius \(r\) whose center is the origin is \(x^{2}+y^{2}=r^{2}\). (a) Use implicit differentiation to determine \(\frac{d y}{d x}\). (b) (Exercise (17), Section 3.2) Let \((a, b)\) be a point on the circle with \(a \neq 0\) and \(b \neq 0\). Determine the slope of the line tangent to the circle at the point \((a, b)\). (c) Prove that the radius of the circle to the point \((a, b)\) is perpendicular to the line tangent to the circle at the point \((a, b)\). Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to -1

Are the following propositions true or false? Justify each conclusion with a counterexample or a proof. (a) For all integers \(a\) and \(b\) with \(a \neq 0,\) the equation \(a x+b=0\) has a rational number solution. (b) For all integers \(a, b,\) and \(c,\) if \(a, b,\) and \(c\) are odd, then the equation \(a x^{2}+b x+c=0\) has no solution that is a rational number. Hint: Do not use the quadratic formula. Use a proof by contradiction and recall that any rational number can be written in the form \(\frac{p}{q},\) where \(p\) and \(q\) are integers, \(q>0\), and \(p\) and \(q\) have no common factor greater than \(1 .\) (c) For all integers \(a, b, c,\) and \(d,\) if \(a, b, c,\) and \(d\) are odd, then the equation \(a x^{3}+b x^{2}+c x+d=0\) has no solution that is a rational number.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.