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Chapter 8: Problem 39

Find the sum of each infinite geometric series. $$3+\frac{3}{4}+\frac{3}{4^{2}}+\frac{3}{4^{3}}+\cdots$$

Short Answer

Expert verified
The sum of the infinite geometric series is 4.

Step by step solution

01

Identify the first term and the ratio

In the series \(3+\frac{3}{4}+\frac{3}{4^{2}}+\frac{3}{4^{3}}+\cdots\), the first term 'a' is 3. The ratio 'r' is the quotient of the second term and the first, that is, \(\frac{3/4}{3} = \frac{1}{4}\).
02

Apply the Infinite Geometric Series Sum Formula

To find the sum of an infinite geometric series, we use the formula \(S = \frac{a}{1-r}\). Substituting \(a = 3\) and \(r = 1/4\) into the formula, we have \(S = \frac{3}{1 - \frac{1}{4}}\).
03

Simplify the Expression

Simplify the denominator by subtracting \(\frac{1}{4}\) from 1 and then divide 3 by this result. So, \(S = \frac{3}{\frac{3}{4}} = 4.\)

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Most popular questions from this chapter

Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is not less than \(5 ?\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sum of the geometric series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{512}\) can only be estimated without knowing precisely which terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}.\)

Solve by the method of your choice. Using 15 flavors of ice cream, how many cones with three different flavors can you create if it is important to you which flavor goes on the top, middle, and bottom?

Make Sense? In Exercises \(78-81,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. The sequence for the number of seats per row in our movie theater as the rows move toward the back is arithmetic with \(d-1\) so people don't block the view of those in the row behind them.

Make Sense? In Exercises \(78-81,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. Beginning at 6: 45 A.M.., a bus stops on my block every 23 minutes, so 1 used the formula for the \(n\) th term of an arithmetic sequence to describe the stopping time for the \(n\) th bus of the day.
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