/*! 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 60 Express each sum using summation... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 60

Express each sum using summation notation. \(1^{3}+2^{3}+3^{3}+\cdots+8^{3}\)

Short Answer

Expert verified
\(\sum_{i=1}^{8} i^3\)

Step by step solution

01

Identify the Pattern

Observe the given sequence: \[1^3 + 2^3 + 3^3 + \ldots + 8^3\]. Each term in the sequence is the cube of an integer.
02

Establish the General Term

Recognize that each term in the sequence can be expressed as \(i^3\) where \(i\) is the integer that ranges from 1 to 8.
03

Use Summation Notation

Express the sequence using the summation notation: \[\sum_{i=1}^{8} i^3\]. This notation indicates that we are summing the cubes of the integers from 1 to 8.

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.

Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In this exercise, we see algebra at play through the use of notation and the manipulation of sequences.
Understanding algebra involves recognizing patterns, establishing general terms, and using symbols to represent mathematical operations.
Here, the sequence of cubes is represented as a series of algebraic expressions, which simplifies the process of understanding and summation.
Sequences and Series
Sequences and series are fundamental concepts in mathematics that deal with ordered lists of numbers and their summation, respectively.
A sequence is simply a set of numbers arranged in a specific order, while a series is the sum of the elements in a sequence.
In the given exercise, the sequence is 1^3, 2^3, 3^3, ..., 8^3. Each term follows a consistent pattern鈥攊t is the cube of consecutive integers from 1 to 8.
This ordered arrangement makes it easier to recognize and express the general term, which in this case, is the cube of any integer within the sequence.
Summation
Summation, often represented by the Greek letter \(\forall\), is the process of adding up a sequence of numbers.
The exercise above illustrates this concept by summing up the cubes of integers from 1 to 8.
Summation notation \(\sum_{i=1}^{8} i^3 \) is used to succinctly represent this process.
It tells us to add the values of \(i^3\) for each integer value of \(i\) from 1 to 8, making it a powerful tool for simplifying complex addition.

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

Droste Effect The Droste Effect, named after the image on boxes of Droste cocoa powder, refers to an image that contains within it a smaller version of the image, which in turn contains an even smaller version, and so on. If each version of the image is \(\frac{1}{5}\) the height of the previous version, the height of the \(n\) th version is given by \(a_{n}=\frac{1}{5} a_{n-1}\). Suppose a Droste image on a package has a height of 4 inches. How tall would the image be in the 6 th version?

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 5\left(\frac{1}{4}\right)^{k-1} $$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the function \(g\) whose graph is the graph of \(y=\sqrt{x}\) but is stretched vertically by a factor of 7 and shifted left 5 units.

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ n^{3}+2 n \text { is divisible by } 3 $$

Show that the statement \(" n^{2}-n+41\) is a prime number" is true for \(n=1\) but is not true for \(n=41\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Probability and 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.