/*! 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 40 Find the sum for each series. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 40

Find the sum for each series. $$\sum_{i=3}^{7}(5 i+2)$$

Short Answer

Expert verified
The sum of the series is 135.

Step by step solution

01

Identify the Series Bounds

The series is represented as \(\sum_{i=3}^{7}(5i+2)\). This indicates that the variable \(i\) starts at 3 and ends at 7. The series will therefore include the terms calculated for \(i = 3, 4, 5, 6,\) and \(7\).
02

Calculate Each Term in the Series

Substitute each value of \(i\) from the bounds into the expression \(5i + 2\).- For \(i = 3\): \(5(3) + 2 = 15 + 2 = 17\)- For \(i = 4\): \(5(4) + 2 = 20 + 2 = 22\)- For \(i = 5\): \(5(5) + 2 = 25 + 2 = 27\)- For \(i = 6\): \(5(6) + 2 = 30 + 2 = 32\)- For \(i = 7\): \(5(7) + 2 = 35 + 2 = 37\)
03

Sum All Terms of the Series

Add all the calculated terms together to find the sum of the series:\(17 + 22 + 27 + 32 + 37 = 135\).

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
Summation is a fundamental concept in algebra and arithmetic, often represented by the symbol \(\Sigma\). It denotes the process of adding a sequence of numbers. In this exercise, the aim is to sum the terms generated by the algebraic expression \(5i + 2\) where \(i\) ranges from 3 to 7. Summation helps us manage and simplify calculations involving numerous terms.
  • The concept of summation is crucial in both pure and applied mathematics.
  • It's used to find the total of a series of numbers.
  • It aids in understanding patterns and results in arithmetic and geometric series.
Always ensure to accurately calculate each term and double-check your added results. This meticulous approach avoids common errors in problem-solving.
Algebraic Expression
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. In our given exercise, the expression \(5i + 2\) is used to generate terms for our series. Algebraic expressions are invaluable tools for summarizing and analyzing mathematical data.
  • An algebraic expression can model real-world situations, making them versatile in mathematics.
  • They involve operations such as addition, subtraction, multiplication, and sometimes powers.
  • When working with expressions, identifying terms and coefficients is vital to understanding their structure and effect.
Understanding how to manipulate these expressions seamlessly integrates into problem-solving in both arithmetic and algebraic contexts.
Finite Series
A finite series is the summation of a sequence that has a clear beginning and end. In this exercise, our series is finite because the variable \(i\) ranges from 3 to 7, giving us a limited number of terms to add. Understanding finite series is essential in many areas of mathematics and helps in analyzing data efficiently.
  • Finite series contrast with infinite series, which continue indefinitely without a clear end point.
  • They are easier to calculate because they involve a specific number of additions.
  • Finite series are used in diverse fields, from statistics to computer science, to solve real-world problems.
Focusing on both the start and end of your series aids in ensuring all terms are correctly included and summed.

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

Use any or all of the methods described in this section to solve each problem. In a club with 8 men and 11 women members, how many 5 -member committees can be chosen that have the following? (a) All men (b) All women (c) 3 men and 2 women (d) No more than 3 women

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{20}=1090, a_{20}=102$$

Prove each statement by mathematical induction. If \(n \geq 4,\) then \(n !>2^{n}\)

Work each problem. \(\quad\) Match each probability in parts (a)-(g) with one of the statements in \(\mathrm{A}-\mathrm{F}\). (a) \(P(E)=-0.1\) (b) \(P(E)=0.01\) (c) \(P(E)=1\) (d) \(P(E)=2\) (e) \(P(E)=0.99\) (f) \(P(E)=0\) (g) \(P(E)=0.5\) A. The event is certain to occur. B. The event is impossible. C. The event is very likely to occur. D. The event is very unlikely to occur. E. The event is just as likely to occur as not to occur. F. This probability cannot occur.

Color-Blind Males The probability that a male will be color blind is \(0.042 .\) Approximate the probabilities that in a group of 53 men, the following are true. A. Exactly 5 are color blind. B. No more than 5 are color blind. C. At least 1 is color blind.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

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.