The Sum Formula for Infinite Geometric Series

Oct 13, 2025 by miodeya.com 66 views

Introduction to Infinite Geometric Series

In mathematics, an infinite geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The series is infinite, meaning it has an infinite number of terms. The sum of an infinite geometric series can be calculated using a specific formula, which is the focus of this article.

The Sum Formula for Infinite Geometric Series

The sum formula for an infinite geometric series is given by:

S_n=\frac{a_1}{1-r}

where:

$S_n$ is the sum of the infinite geometric series
$a_1$ is the first term of the series
$r$ is the common ratio

Understanding the Formula

To understand the formula, let's break it down. The numerator of the fraction is the first term of the series, $a_1$ . The denominator is $1-r$ , where $r$ is the common ratio. The formula states that the sum of the infinite geometric series is equal to the first term divided by $1-r$ .

Applying the Formula to the Given Series

Now, let's apply the formula to the given series:

\sum_{k=0}^{\infty}-20(0.75)^k

In this series, the first term is $-20(0.75)^0 = -20$ . The common ratio is $0.75$ . Plugging these values into the formula, we get:

S_n=\frac{-20}{1-0.75}

Simplifying the Expression

To simplify the expression, we can start by evaluating the denominator:

1-0.75 = 0.25

Now, we can rewrite the expression as:

S_n=\frac{-20}{0.25}

Calculating the Sum

To calculate the sum, we can divide $-20$ by $0.25$ :

S_n = -80

Conclusion

In conclusion, the sum of the infinite geometric series $\sum_{k=0}^{\infty}-20(0.75)^k$ is $-80$ . This can be calculated using the sum formula for infinite geometric series:

S_n=\frac{a_1}{1-r}

where $a_1$ is the first term of the series and $r$ is the common ratio.

Answer Options

Based on the calculation, the correct answer is:

A. -80

The other options are incorrect:

B. $-26 \frac{2}{3}$ is not the correct answer.
C. $26 \frac{2}{3}$ is not the correct answer.
D. 80 is not the correct answer.

Discussion

This problem requires a good understanding of infinite geometric series and the sum formula. The formula is a powerful tool for calculating the sum of an infinite geometric series, and it can be applied to a wide range of problems. In this case, the series is given in a specific form, and the formula can be used to calculate the sum directly.

Real-World Applications

Infinite geometric series have many real-world applications, including:

Finance: Infinite geometric series can be used to model the growth of investments or the decay of debts over time.
Physics: Infinite geometric series can be used to model the behavior of waves or the decay of radioactive materials.
Engineering: Infinite geometric series can be used to model the behavior of electrical circuits or the decay of mechanical systems.

Conclusion

In conclusion, the sum of the infinite geometric series $\sum_{k=0}^{\infty}-20(0.75)^k$ is $-80$ . This can be calculated using the sum formula for infinite geometric series:

S_n=\frac{a_1}{1-r}

where $a_1$ is the first term of the series and $r$ is the common ratio.

Introduction

Infinite geometric series are a fundamental concept in mathematics, and they have many real-world applications. In our previous article, we discussed the sum formula for infinite geometric series and applied it to a specific problem. In this article, we will answer some common questions about infinite geometric series.

Q: What is an infinite geometric series?

A: An infinite geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The series is infinite, meaning it has an infinite number of terms.

Q: What is the sum formula for infinite geometric series?

A: The sum formula for an infinite geometric series is given by:

S_n=\frac{a_1}{1-r}

where:

$S_n$ is the sum of the infinite geometric series
$a_1$ is the first term of the series
$r$ is the common ratio

Q: How do I know if an infinite geometric series converges or diverges?

A: An infinite geometric series converges if the absolute value of the common ratio is less than 1, and it diverges if the absolute value of the common ratio is greater than or equal to 1.

Q: What is the difference between a convergent and a divergent infinite geometric series?

A: A convergent infinite geometric series has a finite sum, while a divergent infinite geometric series has an infinite sum.

Q: How do I calculate the sum of an infinite geometric series?

A: To calculate the sum of an infinite geometric series, you can use the sum formula:

S_n=\frac{a_1}{1-r}

where:

$S_n$ is the sum of the infinite geometric series
$a_1$ is the first term of the series
$r$ is the common ratio

Q: What are some real-world applications of infinite geometric series?

A: Infinite geometric series have many real-world applications, including:

Finance: Infinite geometric series can be used to model the growth of investments or the decay of debts over time.
Physics: Infinite geometric series can be used to model the behavior of waves or the decay of radioactive materials.
Engineering: Infinite geometric series can be used to model the behavior of electrical circuits or the decay of mechanical systems.

Q: Can I use infinite geometric series to model any type of growth or decay?

A: Yes, infinite geometric series can be used to model any type of growth or decay that can be represented by a fixed common ratio.

Q: What are some common mistakes to avoid when working with infinite geometric series?

A: Some common mistakes to avoid when working with infinite geometric series include:

Not checking the convergence of the series: Make sure to check if the absolute value of the common ratio is less than 1 before calculating the sum.
Not using the correct formula: Use the sum formula:

S_n=\frac{a_1}{1-r}

where:

$S_n$ is the sum of the infinite geometric series
$a_1$ is the first term of the series
$r$ is the common ratio

Conclusion

In conclusion, infinite geometric series are a powerful tool for modeling growth and decay in a wide range of fields. By understanding the sum formula and the conditions for convergence, you can use infinite geometric series to solve a variety of problems.

Frequently Asked Questions

Q: What is the sum of the infinite geometric series $\sum_{k=0}^{\infty}-20(0.75)^k$ ? A: The sum of the infinite geometric series $\sum_{k=0}^{\infty}-20(0.75)^k$ is $-80$ .
Q: How do I know if an infinite geometric series converges or diverges? A: An infinite geometric series converges if the absolute value of the common ratio is less than 1, and it diverges if the absolute value of the common ratio is greater than or equal to 1.
Q: What are some real-world applications of infinite geometric series? A: Infinite geometric series have many real-world applications, including finance, physics, and engineering.

Additional Resources

Infinite Geometric Series Calculator: Use this calculator to calculate the sum of an infinite geometric series.
Infinite Geometric Series Formula: Use this formula to calculate the sum of an infinite geometric series:

S_n=\frac{a_1}{1-r}

where:

$S_n$ is the sum of the infinite geometric series
$a_1$ is the first term of the series
$r$ is the common ratio