/*! 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 65 Using Summation Notation Use sum... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 65

Using Summation Notation Use summation notation to write the sum. $$5+15+45+\cdots+3645$$

Short Answer

Expert verified
The sum of the series \(5+15+45+...+3645\) in summation (sigma) notation is \(\Sigma_{n=1}^{8} 5 \times 3^{n-1}\)

Step by step solution

01

Identify the pattern

Examining the series, it can be noticed that each term is three times the term before it. All terms can be represented using the formula \(5 \times 3^{n-1}\) where n starts at 1 and each subsequent term increments n by 1.
02

Find the number of terms

With the series starting at 5 and ending at 3645, and since each term is thrice the previous term, there are 8 terms in total.
03

Write in Sigma notation

The sigma notation of the series \(5+15+45+...+3645\) is \(\Sigma_{n=1}^{8} 5 \times 3^{n-1}\)

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.

Geometric Series
A geometric series is a sum of terms that have a constant ratio between consecutive terms. This ratio is known as the common ratio. For the series in the problem, each term is obtained by multiplying the previous term by 3. This identifies the series as a geometric progression with a common ratio of 3.
Key points to understand about geometric series include:
  • The first term is the starting point, which in this series is 5.
  • The common ratio, which tells us how each term relates to its predecessor, is 3 in this case.
  • The formula for the n-th term is generally given by \(a \cdot r^{(n-1)}\), where \(a\) is the first term and \(r\) is the common ratio.
Geometric series are fundamental in mathematics due to their simple yet powerful structure, making them a key tool for solving many types of problems involving exponential growth or decay.
Understanding these concepts helps in identifying series patterns and efficiently solving related problems.
Sigma Notation
Sigma notation is a compact and powerful way to express the sum of a sequence of terms. It is denoted by the Greek letter \(\Sigma\), which stands for "sum."
In the problem, the series \(5+15+45+\cdots+3645\) is expressed using sigma notation as \(\Sigma_{n=1}^{8} 5 \times 3^{n-1}\). Here's what each part represents:
  • The \(n=1\) below the \(\Sigma\) symbol signifies the starting index for the series.
  • The superscript 8 above the \(\Sigma\) symbol is the ending index, indicating there are 8 total terms.
  • The expression \(5 \times 3^{n-1}\) is the formula for each term in the series, describing how each term is calculated based on \(n\).
Sigma notation makes it easier to handle and manipulate large sums, especially when associated with geometric series or other patterns. It's commonly used in calculus and higher mathematics to deal with infinite series or sums of functions over intervals.
Series Representation
The representation of a series involves expressing it in a form that is concise yet informative, allowing one to easily grasp the nature of the sequence.
For example, in the original exercise, the series \(5 + 15 + 45 + \cdots + 3645\) is represented concisely in the step-by-step solution using sigma notation. This is important for several reasons:
  • It provides a clear view of the terms involved and the pattern they follow.
  • Facilitates the computation of the sum by providing a direct formula for each term.
  • Simplifies the process of deducing properties of the series, such as the number of terms and the progression pattern.
Representing series efficiently is crucial for simplifying complex expressions and performing mathematical operations on them. It serves as an essential skill in mathematical problem-solving, making the process more manageable and intuitive for learners.

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

Write the first five terms of the sequence defined recursively. Use the pattern to write the \(n\) th term of the sequence as a function of \(n .\) (Assume \(n\) begins with 1.) $$a_{1}=25, a_{k+1}=a_{k}-5$$

Write the first five terms of the sequence defined recursively. $$a_{1}=32, a_{k+1}=\frac{1}{2} a_{k}$$

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 0.) $$a_{n}=\frac{1}{n !}$$

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$9,-6,4,-\frac{8}{3}, \dots$$

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$4,7,12,19,28, \dots$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.