/*! 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 98 Find the sum. $$\sum_{j=0}^{4}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 98

Find the sum. $$\sum_{j=0}^{4}(-2)^{j}$$

Short Answer

Expert verified
The sum of the geometric sequence is -11.

Step by step solution

01

Identify sequence type

You must identify the type of sequence or series. Here, the sum notation and the power of j attached to -2 indicates a geometric sequence. In a geometric sequence each term after the first is found by multiplying the previous term by a fixed, non-zero number. In this case, that number is -2.
02

Determine the parameters

Our series starts at j=0 and ends at j=4. The common ratio (r) is -2, which is the number being raised to the power of j. The number of terms (n) is 5 as our count starts from 0.
03

Use formula to find the sum

The formula to find the sum of a geometric sequence is \(S = \frac{a(r^n - 1)}{r - 1}\) where 'a' is the first term, 'r' is the common ratio and 'n' is the total number of terms. But since our series starts from 0, it essentially means the first term of our series is 1 because -2^0 = 1. So, applying the values to the formula, we get: \(S = \frac{1((-2)^5 - 1)}{-2 - 1}\).
04

Simplify the equation

Now simplify the equation: \(S = \frac{1(-32 - 1)}{-3} = \frac{33}{3}= -11\)

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

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=\frac{3^{n}}{3^{n}+1}\\\ &a_{6}= \end{aligned}$$

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.) $$a_{n}=\frac{(-1)^{2 n+1}}{(2 n+1) !}$$

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{250}(1000-n)$$

About It The sum of the first \(n\) terms of an arithmetic sequence with first term \(a_{1}\) and common difference \(d\) is \(S_{n} .\) Determine the sum when each term is increased by \(5 .\) Explain.

An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on (see figure). How many seats are there in all 20 rows?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.