/*! 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 2 Evaluate the finite series for t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 2

Evaluate the finite series for the specified number of terms. $$ 4+12+36+\ldots ; n=6 $$

Short Answer

Expert verified
The sum of the provided series up to 6 terms is -1456.

Step by step solution

01

Identify the Common Ratio and the First Term

The given series is a geometric series where each term is three times the previous term, so the common ratio \( r \) is 3. The first term \( a \) is 4.
02

Apply the Formula for the Sum of a Geometric Series

The formula for the sum \( S \) of the geometric series to \( n \) terms is: \( S = a \cdot \frac{1 - r^n}{1 - r} \)
03

Substitute the Values

Substitute \( a = 4 \), \( r = 3 \), and \( n = 6 \) into the formula: \( S = 4 \cdot \frac{1 - 3^6}{1 - 3} \)
04

Calculate the Sum

Simplifying the expression results in the sum of the series: \( S = 4 \cdot \frac{1 - 729}{-2} = -1456 \)

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.

Common Ratio
In a geometric series, the common ratio is the fixed number you multiply by to get from one term to the next. Identifying this ratio is crucial in understanding how the series progresses.
In our example, the series is 4, 12, 36, and so on. Each term is obtained by multiplying the preceding term by 3. Thus, the common ratio, denoted by the symbol \( r \), is 3.
This pattern is consistent:
  • The second term (12) is 4 times 3.
  • The third term (36) is 12 times 3.
This consistent multiplication is what defines the geometric nature of the series. Understanding the common ratio allows you to predict and continue the sequence indefinitely if needed.
Sum of a Series
Calculating the sum of a geometric series, especially when dealing with a finite number of terms, follows a specific formula.
The formula is:
\[ S = a \cdot \frac{1 - r^n}{1 - r} \]
Here, \( S \) represents the sum, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms in the series.
This formula efficiently finds the sum without manually adding each term. In our problem, for instance, substituting \( a = 4 \), \( r = 3 \), and \( n = 6 \) into the formula gives us \( S = 4 \times \frac{1 - 3^6}{1 - 3} \). This simplifies to \( S = -1456 \), giving the complete sum of the series in just a few calculations.
Finite Series
A finite series means there is a set number of terms. This is opposed to an infinite series, which continues without end.
In the given problem, the series is finite because it specifies 6 terms. Calculating sums of finite series is often simpler and more achievable since you know precisely how many terms you'll need.
Finite series are common in practical applications. They help determine things like total costs over fixed periods or total growth efforts. Recognizing that our series is finite simplifies calculations, as it provides an endpoint where calculations can stop.
First Term
The first term of a series is the initial value from which the series starts. It is denoted as \( a \).
In our example series, the first term is 4. This term sets the starting point for calculating the subsequent terms in the series using the common ratio.
Knowing the first term is essential for applying the sum formula for a geometric series since it helps set up the series' initial conditions. Without it, you couldn't effectively compute or understand the growing pattern of the series.

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

critical Thinking Use your graphing calculator to find the area of the triangle with vertices \((-3,0),(-1,3),\) and \((1,0) .\) (Hint. First find the function whose graph makes a peak at \((-1,3) . )\)

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms. \(2+4+8+16+\ldots ; n=10\)

Evaluate each infinite geometric series. $$ 3+1+\frac{1}{3}+\frac{1}{9}+\ldots $$

Write and evaluate a sum to estimate the area under each curve for the domain \(0 \leq x \leq 2\) . a. Use inscribed rectangles 1 unit wide. b. Use eircumscribed rectangles 1 unit wide. $$ h(x)=-(x-2)^{2}+5 $$

Solve each equation. Check your solution. $$ \frac{5}{2-x}=\frac{4}{2 x+1} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

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.