/*! 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 28 Evaluate the geometric series. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 28

Evaluate the geometric series. $$ \sum_{k=1}^{90} \frac{5}{7^{k}} $$

Short Answer

Expert verified
The sum of the geometric series is given by \( S = \frac{35}{6}(1-(\frac{1}{7})^{90}) \).

Step by step solution

01

Find the first term (a)

The first term, \(a\), is simply the value of the series when k=1. So, when \(k=1\): $$ a=\frac{5}{7^{1}} = \frac{5}{7} $$
02

Find the common ratio (r)

The common ratio, \(r\), is the factor at which each term multiplies to get the next term. In this case: $$ r=\frac{1}{7} $$
03

Find the number of terms (n)

The number of terms, \(n\), is simply the difference between the upper and lower bounds of the summation plus 1. So in this case: $$ n = 90 - 1 +1 = 90 $$
04

Apply the sum formula

Now that we have the values for \(a\), \(r\), and \(n\), we can apply the formula to evaluate the sum of the geometric series: $$ S = \frac{a(1-r^n)}{1-r} = \frac{\frac{5}{7}(1-(\frac{1}{7})^{90})}{1-\frac{1}{7}} $$
05

Simplify the expression

Now we simplify the expression: $$ S = \frac{\frac{5}{7}(1-(\frac{1}{7})^{90})}{\frac{6}{7}} $$ Next, we can cancel out the common factors in the numerator and the denominator: $$ S = 5(1-(\frac{1}{7})^{90}) \cdot \frac{7}{6} $$
06

Final answer

Finally, we get the sum of the geometric series: $$ S = \frac{35}{6}(1-(\frac{1}{7})^{90}) $$

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!

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

Evaluate \(\lim _{n \rightarrow \infty} \frac{2 n^{2}+5 n+1}{5 n^{2}-6 n+3}\).

Evaluate the geometric series. $$ \frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots+\frac{1}{4^{50}} $$

Evaluate the arithmetic series. $$ 300+293+286+\cdots+55+48+41 $$

Assume \(n\) is a positive integer. Evaluate \(\left(\begin{array}{c}n \\ n-1\end{array}\right)\).

Restate the symbolic version of the formula for evaluating a geometric series using summation notation.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.