/*! 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 61 Write out the terms of the serie... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 61

Write out the terms of the series and then evaluate it. $$\sum_{k=1}^{4}(k+1)$$

Short Answer

Expert verified
The sum of the series is 14.

Step by step solution

01

Write Out the Series Terms

The expression for each term in the series is given by \(k+1\). We need to calculate this expression for each value of \(k\) from 1 to 4 and list those terms. Substituting the values: - For \(k = 1\), the term is \(1+1 = 2\).- For \(k = 2\), the term is \(2+1 = 3\).- For \(k = 3\), the term is \(3+1 = 4\).- For \(k = 4\), the term is \(4+1 = 5\).
02

List the Series

Now, list the terms we found from Step 1 in sequence: 2, 3, 4, 5. This forms the series that results from the summation notation \( \, \sum_{k=1}^{4}(k+1) \, \).
03

Evaluate the Sum of the Series

Add the terms of the series together to evaluate the sum:- Start with the first two terms: \(2 + 3 = 5\). - Add this result to the next term: \(5 + 4 = 9\).- Finally, add the last term: \(9 + 5 = 14\). Thus, the sum of the series is 14.

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.

Summation Notation
Summation notation, also known as sigma notation, is a convenient way to express the sum of a sequence of numbers. It is represented by the Greek letter sigma (\( \Sigma \)). This notation is especially helpful when dealing with a large number of terms or complex series.

In the given exercise, the summation notation \( \sum_{k=1}^{4}(k+1) \) tells us to sum up the expression \(k + 1\) for values of \(k\) starting from 1 going up to 4.
  • The lower limit (1) indicates where to start.
  • The upper limit (4) tells where to stop.
  • The expression \(k + 1\) inside the sigma gives the rule for finding each term.
Recognizing and understanding the components of sigma notation allows you to quickly set up and calculate a series.
Arithmetic Series
An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference.

In our series \(2, 3, 4, 5\) derived from \(k + 1\), you can see that each number increases by 1 from the previous number. This makes it an arithmetic progression, because:
  • The first term is 2.
  • The common difference between terms is 1 (since \( 3-2 = 1, 4-3 = 1, \) and \( 5-4 = 1\)).
The series derived using sigma notation simplifies to understanding how each number builds off the last. Recognizing a sequence as arithmetic simplifies prediction and calculation of terms.
Finite Series
A finite series is one that has a definite number of terms. In contrast to an infinite series, which continues indefinitely, a finite series has a clear start and end.
  • The series \(2, 3, 4, 5\) derived from \(\sum_{k=1}^{4}(k+1)\) is finite as it includes only terms from \(k=1\) to \(k=4\).
  • Finite series allow for simple calculation of the total sum, as you just add up all terms listed.
Evaluating a finite series, as shown in the solution, involves listing out each term and summing them up, resulting in the sum of 14 for our example.

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

Suppose that the density of female insects during the first year is 500 per acre with \(r=0.8\). (a) Write a recursive sequence that describes these data, where \(a_{n}\) denotes the female insect density during year \(n\). (b) Find the six terms \(a_{1}, a_{2}, a_{3}, \ldots, a_{6}\). Interpret the results. (c) Find a formula for \(a_{n}\).

The following recursively defined sequence can be used to compute \(\sqrt{k}\) for any positive number \(k .\) \(a_{1}=k ; a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right)\) This sequence was known to Sumerian mathematicians 4000 years ago, but it is still used today. Use this sequence to approximate the given square root by finding a \(6 .\) Compare your result with the actual value. $$\sqrt{21}$$

Use Pascal's triangle to help expand the expression. $$ (m+n)^{3} $$

Find the probability of rolling a die five times and obtaining a 6 on the first two rolls, a 5 on the third roll, and a \(1,2,3,\) or 4 on the last two rolls.

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$-5,2,9,16,23,30$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.