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Chapter 11: Problem 40

Find the sum of each infinite geometric series. $$ 5+\frac{5}{6}+\frac{5}{6^{2}}+\frac{5}{6^{3}}+\cdots $$

Short Answer

Expert verified
The sum of this infinite geometric series is 6.

Step by step solution

01

Identify the first term

The first term \(a\) of the series is the first number of the series, which is 5.
02

Identify the common ratio

The common ratio \(r\) is found by dividing any term in the sequence by the previous term. Here, it is \(\frac{5}{6}\) divided by 5 which is \(\frac{1}{6}\).
03

Apply the formula

Now we can apply the formula for the sum of an infinite geometric series: \(S=\frac{a}{1-r}\). Substituting the values a=5, r=\(\frac{1}{6}\) into the formula, we get \(S=\frac{5}{1-\frac{1}{6}}\)
04

Simplify the expression

Simplifying the expression gives \(S=\frac{5}{\frac{5}{6}}\). Thus, S=6

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Sequence
A geometric sequence is a list of numbers where each term is obtained by multiplying the previous one by a constant, known as the common ratio. In our exercise, the sequence begins with 5 and continues indefinitely. Each term follows this pattern:
  • The second term is \( \frac{5}{6} \).
  • The third term is \( \frac{5}{6^2} \).
  • The fourth term is \( \frac{5}{6^3} \).
This regular pattern continues forever.
The geometric nature means a consistent multiplication process at work, distinguishing it from an arithmetic sequence where addition is the rule. Instead, in a geometric sequence, the multiplicative nature drives the progression of terms.
Sum of Series
The sum of an infinite geometric series depends on the series' properties, specifically having a common ratio smaller than 1 in absolute value. This ensures the terms diminish, allowing the total sum to converge to a finite limit.
In the given series, using the sum formula, the convergence means no matter how many terms you add, the series approaches a precise number, not infinity. The sum of our infinite series using the formula \( S = \frac{a}{1-r} \) yields a clean result, ensuring convergence and allowing practical calculation.
Common Ratio
The common ratio is the backbone of any geometric sequence, determining how each term relates to the one before it. Calculated by dividing any term by its predecessor, it remains consistent across the sequence.
In the example, dividing \( \frac{5}{6} \) by 5 results in \( \frac{1}{6} \). This consistent pattern facilitates the sequential multiplication that defines geometric sequences. Because this ratio is less than 1, it ensures that each subsequent term in our series decreases in magnitude, a crucial factor allowing the series to converge to a sum.
First Term
The first term in a geometric series is the starting point from which all other terms are derived. It sets the series in motion and is represented symbolically by \( a \).
In our problem, the first term is straightforwardly identified as 5, the initial term given.
This value is essential in applying formulas like the sum of the series formula, \( S = \frac{a}{1-r} \), which requires knowing the starting point to calculate the total sum accurately. It serves as the base of the sequence from which every subsequent term is generated, relying heavily on the common ratio.

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Most popular questions from this chapter

The table shows the population of California for 2000 and \(2010,\) with estimates given by the U.S. Census Bureau for 2001 through 2009 \(\begin{array}{lllllll}\hline \text { Year } & {2000} & {2001} & {2002} & {2003} & {2004} & {2005} \\ \hline \text { Population } & {33.87} & {34.21} & {34.55} & {34.90} & {35.25} & {35.60} \\ \hline\end{array}\) \(\begin{array}{llllll}{\text { Year }} & {2006} & {2007} & {2008} & {2009} & {2010} \\ {\text { Population }} & {36.00} & {36.36} & {36.72} & {37.09} & {37.25}\end{array}\) a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after 1999 c. Use your model from part (b) to project California's population, in millions, for the year \(2020 .\) Round to two decimal places.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I toss a coin, the probability of getting heads or tails is 1 but the probability of getting heads and tails is \(0 .\)

If you toss a fair coin six times, what is the probability of getting all heads?

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\) In Exercises \(71-72,\) you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, continuing to double your savings each day. What will your total savings be for the first 30 days?

Explaining the Concepts What are mutually exclusive events? Give an example of two events that are mutually exclusive.
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