/*! 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 45 Write each series using summatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 45

Write each series using summation notation. $$ 3+4+5+6+7 $$

Short Answer

Expert verified
Summation Notation: \[ \sum_{n=1}^{5} (3 + (n-1)) \]

Step by step solution

01

Identify the Pattern

Look at the series: 3, 4, 5, 6, 7. Observe that each term increases by 1.
02

Determine the General Term

The first term is 3 and each subsequent term increases by 1. Therefore, the n-th term can be written as: \[ a_n = 3 + (n-1) \]
03

Identify the Range of the Index

The series starts at 3 (when the index is 1) and ends at 7 (when the index is 5). So, the index n ranges from 1 to 5.
04

Write in Summation Notation

Combine the general term and the range of the index to write the series using summation notation: \[ \sum_{n=1}^{5} (3 + (n-1)) \]

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.

Series
A series is the sum of the terms of a sequence. In this exercise, the given series is 3, 4, 5, 6, 7. Identifying a pattern in the series is crucial to understand how to express it in summation notation. With each element increasing by 1, we can use this pattern to formulate a more general expression for the sum. Recognizing the structure of the series helps you convert it into a concise mathematical representation that is easier to work with.
General Term
The general term of a series provides a formula to determine any term in the sequence based on its position. For the series 3, 4, 5, 6, 7, you can see that the first term is 3 and subsequent terms increase by 1. Representing this general trend mathematically, we get:
  • First term ( n = 1): 3 + (1-1) = 3
  • Second term ( n = 2): 3 + (2-1) = 4
  • Third term ( n = 3): 3 + (3-1) = 5
Thus, the general term for this series can be written as: \[a_n = 3 + (n-1)\] This allows you to plug in any value of n to find the corresponding term in the series.
Index Range
In summation notation, the index range defines the start and end points of the summation. For the series 3, 4, 5, 6, 7, the index starts at 1 and ends at 5. This means:
  • When n = 1, the term is 3
  • When n = 2, the term is 4
  • When n = 3, the term is 5
  • When n = 4, the term is 6
  • When n = 5, the term is 7
Therefore, we can indicate the index range using n = 1 to n = 5. This range helps provide clarity on the terms included in the series.
Algebra
Using algebraic manipulation, we can convert a series into a summation notation. Starting with the general term:
  • \[a_n = 3 + (n-1)\]
we combine this with the index range from 1 to 5: \[\sum_{n=1}^{5} (3 + (n-1))\] This equation represents the sum of the series from the first term to the fifth term. Algebra allows us to simplify expressions and helps in performing operations like summation in an efficient manner. Understanding these transformations is essential for solving series-related problems.

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 mathematical induction to prove that each statement is true for every positive integer value of \(n.\) $$\text { If } a>1, \text { then } a^{n}>a^{n-1}$$

Use mathematical induction to prove that each statement is true for every positive integer value of \(n.\) $$x^{2 n}+x^{2 n-1} y+\cdots+x y^{2 n-1}+y^{2 n}=\frac{x^{2 n+1}-y^{2 n+1}}{x-y}$$

Use mathematical induction to prove that each statement is true for every positive integer value of \(n.\) $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$

Write each series as a sum of terms and then find the sum. $$ \sum_{i=1}^{4}\left(i^{3}+3\right) $$

Give answers to the nearest thousandth. $$ a_{1}=-3, r=4 ; \quad \text { Find } S_{10}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.