/*! 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 3 Prove \(1^{3}+2^{3}+\dots+n^{3}=... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Prove \(1^{3}+2^{3}+\dots+n^{3}=(1+2+\dots+n)^{2}\) for all natural numbers \(\boldsymbol{n}\)

Short Answer

Expert verified
We used the formulas for the sum of the first n natural numbers and their cubes to show the equality.

Step by step solution

01

Identify the formula to prove

We need to prove that the sum of cubes of the first n natural numbers is equal to the square of the sum of the first n natural numbers. In symbols, we need to show that: \[ 1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2 \]
02

Recall the formula for the sum of the first n natural numbers

The sum of the first n natural numbers is given by the formula: \[ 1 + 2 + \cdots + n = \frac{n(n+1)}{2} \]
03

Square the sum of the first n natural numbers

We now square both sides of the sum of the first n natural numbers: \[ \left( \frac{n(n+1)}{2} \right)^2 = \frac{n^2(n+1)^2}{4} \]
04

Use the formula for the sum of cubes of the first n natural numbers

The sum of the first n cubes is given by the formula: \[ 1^3 + 2^3 + \cdots + n^3 = \left( \frac{n(n+1)}{2} \right)^2 \]
05

Conclusion

Since both expressions, \( 1^3 + 2^3 + \cdots + n^3 \) and \( \left( \frac{n(n+1)}{2} \right)^2 \), match, we have proved that: \[ 1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2 \]

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.

Natural Numbers
Natural numbers are the set of positive integers starting from 1. They are represented as \(\text{N} = \{1, 2, 3, 4, 5, \ldots\}\). Natural numbers are countable and are fundamental to number theory. They form the basis of the arithmetic we use every day. In this context, we are dealing with sums of natural numbers and their properties. Understanding natural numbers helps in defining sequences like the sum of cubes and the sum of squares.
Sum of Squares
The sum of squares for the first n natural numbers is another interesting sequence. It is given by the formula: \[ 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6} \] This formula helps in understanding several properties of numbers in sequences and is essential when exploring various types of sums and patterns in mathematics. It is related, but different from the sum of cubes, and serves as a stepping stone toward more complex identities.
Mathematical Induction
Mathematical induction is a powerful technique used to prove statements for all natural numbers. The process involves two steps:
  • Base Case: Verify the statement for the initial value, usually n = 1.
  • Inductive Step: Assume the statement is true for some arbitrary value n = k, then prove it is true for n = k + 1.
By following these steps, you can prove that a statement holds for all natural numbers. In our context, mathematical induction can be used to solidify the understanding of why the sum of cubes equals the square of the sum of natural numbers.
Algebraic Identities
Algebraic identities are crucial tools in simplifying and proving various mathematical formulas. An identity provides an equation that is true for all values of the variables within a certain range. For example, the identity we are dealing with in this exercise is: \[ 1^3 + 2^3 + \ldots + n^3 = \left(1 + 2 + \ldots + n\right)^2 \] and its related sum formulas. Such identities help in simplifying complex algebraic expressions and proving important mathematical theorems. Mastering these identities can significantly enhance your problem-solving skills.

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

Show that \(2^{1 / 3}, 5^{1 / 7},\) and \((13)^{1 / 4}\) do not represent rational numbers.

(a) Prove that \(|a+b+c| \leq|a|+|b|+|c|\) for all \(a, b, c \in\) R. Hint: Apply the triangle inequality twice. Do not consider eight cases. (b) Use induction to prove $$\left|a_{1}+a_{2}+\dots+a_{n}\right| \leq\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{n}\right| $$for \(n\) numbers \(a_{1}, a_{2}, \ldots, a_{n}\)

Prove \(1^{2}+2^{2}+\dots+n^{2}=\frac{1}{6} n(n+1)(2 n+1)\) for all natural numbers \(n\)

(a) Show that \(\left\\{r \in \mathbb{Q}: r^{3}<2\right\\}\) is a Dedekind cut, but that \(\\{r \in \mathbb{Q}:\) \(\left.r^{2}<2\right\\}\) is not a Dedekind cut. (b) Does the Dedekind cut \(\left\\{r \in \mathbb{Q}: r^{3}<2\right\\}\) correspond to a rational number in \(\mathbb{R} ?\) (c) Show that \(0^{*} \cup\left\\{r \in \mathbb{Q}: r \geq 0 \text { and } r^{2}<2\right\\}\) is a Dedekind cut. Does it correspond to a rational number in \(\mathbb{R} ?\)

Show that if \(\alpha\) and \(\beta\) are Dedekind cuts, then so is \(\alpha+\beta=\left\\{r_{1}+r_{2}:\right.\) \(\left.r_{1} \in \alpha \text { and } r_{2} \in \beta\right\\}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.