Sum of an Infinite Geometric Series

Oct 13, 2025 by leolavanderia.com.br 63 views

Introduction

Imagine a painter tasked with painting a wall with an area of $$150 , \text{ft}^2$$. He decides to paint half of the wall and then take a break. After his break, he paints half of the remaining unpainted portion and then takes another break. If he continues this pattern, painting half of the remaining wall with each break, we are left with a seemingly infinite series of fractions. In this article, we will delve into the world of mathematics and explore the concept of infinite series, specifically the geometric series, to understand the painter's progress.

The Painter's Progress

Let's assume the wall has an area of $$150 , \text{ft}^2$$. The painter starts by painting half of the wall, which is $$\frac{1}{2} \times 150 = 75 , \text{ft}^2$$. After his first break, he has painted $$75 , \text{ft}^2$$ and has $$75 , \text{ft}^2$$ left to paint.

After his break, he paints half of the remaining unpainted portion, which is $$\frac{1}{2} \times 75 = 37.5 , \text{ft}^2$$. Now, he has painted a total of $$75 + 37.5 = 112.5 , \text{ft}^2$$ and has $$37.5 , \text{ft}^2$$ left to paint.

This pattern continues, with the painter painting half of the remaining unpainted portion with each break. We can represent this as an infinite series:

\frac{1}{2} \times 150 + \frac{1}{2} \times \frac{1}{2} \times 150 + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times 150 + \ldots

Geometric Series

The series we have is a geometric series, where each term is obtained by multiplying the previous term by a fixed constant, in this case, $$\frac{1}{2}$$. A geometric series can be represented as:

a + ar + ar^2 + ar^3 + \ldots

where $$a$$ is the first term and $$r$$ is the common ratio.

In our case, $$a = \frac{1}{2} \times 150 = 75$$ and $$r = \frac{1}{2}$$.

Sum of an Infinite Geometric Series

The sum of an infinite geometric series can be calculated using the formula:

S = \frac{a}{1 - r}

where $$S$$ is the sum, $$a$$ is the first term, and $$r$$ is the common ratio.

Plugging in the values, we get:

S = \frac{75}{1 - \frac{1}{2}} = \frac{75}{\frac{1}{2}} = 150

The Painter's Progress Revisited

Now that we have the sum of the infinite series, we can revisit the painter's progress. The painter starts by painting half of the wall, which is $$\frac{1}{2} \times 150 = 75 , \text{ft}^2$$. After his first break, he has painted $$75 , \text{ft}^2$$ and has $$75 , \text{ft}^2$$ left to paint.

This pattern continues, with the painter painting half of the remaining unpainted portion with each break. We can represent this as an infinite series:

\frac{1}{2} \times 150 + \frac{1}{2} \times \frac{1}{2} \times 150 + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times 150 + \ldots

The sum of this infinite series is $$150 , \text{ft}^2$$, which means that the painter will eventually paint the entire wall.

Conclusion

In this article, we explored the concept of infinite series, specifically the geometric series, to understand the painter's progress. We represented the painter's progress as an infinite series and calculated the sum of the series using the formula for the sum of an infinite geometric series. We found that the sum of the series is $$150 , \text{ft}^2$$, which means that the painter will eventually paint the entire wall.

References

[1] "Geometric Series" by Math Open Reference. Retrieved from https://www.mathopenref.com/seriesgeometric.html
[2] "Infinite Series" by Khan Academy. Retrieved from https://www.khanacademy.org/math/infinite-series

Introduction

In our previous article, we explored the concept of infinite series, specifically the geometric series, to understand the painter's progress. We represented the painter's progress as an infinite series and calculated the sum of the series using the formula for the sum of an infinite geometric series. In this article, we will answer some of the most frequently asked questions about the painter's infinite series.

Q: What is the painter's infinite series?

A: The painter's infinite series is a geometric series that represents the painter's progress as he paints half of the remaining unpainted portion of the wall with each break.

Q: How is the painter's infinite series represented mathematically?

A: The painter's infinite series can be represented mathematically as:

\frac{1}{2} \times 150 + \frac{1}{2} \times \frac{1}{2} \times 150 + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times 150 + \ldots

Q: What is the sum of the painter's infinite series?

A: The sum of the painter's infinite series is $$150 , \text{ft}^2$$, which means that the painter will eventually paint the entire wall.

Q: How does the painter's infinite series relate to the concept of infinite series?

A: The painter's infinite series is a classic example of an infinite series, specifically a geometric series. It demonstrates how an infinite series can be used to model real-world problems and how the sum of an infinite series can be calculated using the formula for the sum of an infinite geometric series.

Q: What are some real-world applications of the painter's infinite series?

A: The painter's infinite series has several real-world applications, including:

Modeling population growth
Calculating the sum of an infinite geometric series
Understanding the concept of infinite series

Q: Can the painter's infinite series be used to model other real-world problems?

A: Yes, the painter's infinite series can be used to model other real-world problems, such as:

Modeling the growth of a population
Calculating the sum of an infinite geometric series
Understanding the concept of infinite series

Q: What are some common misconceptions about the painter's infinite series?

A: Some common misconceptions about the painter's infinite series include:

Thinking that the painter will never finish painting the wall
Thinking that the sum of the series is infinite
Thinking that the painter's infinite series is a unique problem

Q: How can the painter's infinite series be used to teach mathematical concepts?

A: The painter's infinite series can be used to teach mathematical concepts, such as:

The concept of infinite series
The formula for the sum of an infinite geometric series
The concept of convergence and divergence

Q: What are some resources for learning more about the painter's infinite series?

A: Some resources for learning more about the painter's infinite series include:

Online tutorials and videos
Mathematical textbooks and resources
Online forums and communities

Conclusion

In this article, we answered some of the most frequently asked questions about the painter's infinite series. We hope that this article has provided a better understanding of the concept of infinite series and how it can be used to model real-world problems.

References

[1] "Geometric Series" by Math Open Reference. Retrieved from https://www.mathopenref.com/seriesgeometric.html
[2] "Infinite Series" by Khan Academy. Retrieved from https://www.khanacademy.org/math/infinite-series
[3] "The Painter's Dilemma" by The Math Forum. Retrieved from https://mathforum.org/library/drmath/view/65651.html
[4] "The Infinite Series Problem" by The Wolfram Demonstrations Project. Retrieved from https://demonstrations.wolfram.com/TheInfiniteSeriesProblem/