Geometric Series Applications

Oct 13, 2025 by leolavanderia.com.br 108 views

Introduction

In mathematics, a geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant. The formula for a geometric series is given by $\sum_{n=1}^{\infty} ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio. In this article, we will focus on a specific geometric series represented by $\sum_{n=1}^6(8)\left(\frac{1}{2}\right)^{n-1}$ and calculate its sum.

Understanding the Formula

The given series is a geometric series with the first term $a = 8$ and the common ratio $r = \frac{1}{2}$ . The series is finite, meaning it has a limited number of terms, and it is given by $\sum_{n=1}^6(8)\left(\frac{1}{2}\right)^{n-1}$ . To calculate the sum of this series, we can use the formula for a finite geometric series, which is given by:

S_n = \frac{a(1-r^n)}{1-r}

where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Calculating the Sum

To calculate the sum of the given series, we can plug in the values of $a$ , $r$ , and $n$ into the formula for a finite geometric series. We have $a = 8$ , $r = \frac{1}{2}$ , and $n = 6$ . Substituting these values into the formula, we get:

S_6 = \frac{8\left(1-\left(\frac{1}{2}\right)^6\right)}{1-\frac{1}{2}}

Simplifying the expression, we get:

S_6 = \frac{8\left(1-\frac{1}{64}\right)}{\frac{1}{2}}

S_6 = \frac{8\left(\frac{63}{64}\right)}{\frac{1}{2}}

S_6 = 16\left(\frac{63}{64}\right)

S_6 = \frac{1008}{64}

S_6 = 15.75

Conclusion

In this article, we have calculated the sum of a geometric series represented by $\sum_{n=1}^6(8)\left(\frac{1}{2}\right)^{n-1}$ . We used the formula for a finite geometric series to calculate the sum, and we obtained a value of 15.75. This result demonstrates the power of geometric series in mathematics and their applications in various fields.

Geometric Series Formula

The formula for a geometric series is given by $\sum_{n=1}^{\infty} ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio. The formula for a finite geometric series is given by:

S_n = \frac{a(1-r^n)}{1-r}

where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Common Ratio

The common ratio $r$ is a fixed constant that is used to obtain each term in the series. In the given series, the common ratio is $\frac{1}{2}$ .

First Term

The first term $a$ is the first term in the series. In the given series, the first term is $8$ .

Number of Terms

The number of terms $n$ is the number of terms in the series. In the given series, the number of terms is $6$ .

Geometric Series Applications

Geometric series have numerous applications in mathematics and other fields. Some of the applications include:

Finance: Geometric series are used to calculate the future value of an investment.
Engineering: Geometric series are used to calculate the stress and strain on a material.
Computer Science: Geometric series are used to calculate the time complexity of an algorithm.

Conclusion

In conclusion, geometric series are a powerful tool in mathematics and have numerous applications in various fields. The formula for a geometric series is given by $\sum_{n=1}^{\infty} ar^{n-1}$ , and the formula for a finite geometric series is given by:

S_n = \frac{a(1-r^n)}{1-r}

Q: What is a geometric series?

A: A geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant. The formula for a geometric series is given by $\sum_{n=1}^{\infty} ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.

Q: What is the formula for a geometric series?

A: The formula for a geometric series is given by $\sum_{n=1}^{\infty} ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.

Q: What is the formula for a finite geometric series?

A: The formula for a finite geometric series is given by:

S_n = \frac{a(1-r^n)}{1-r}

where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: What is the common ratio?

A: The common ratio $r$ is a fixed constant that is used to obtain each term in the series.

Q: What is the first term?

A: The first term $a$ is the first term in the series.

Q: What is the number of terms?

A: The number of terms $n$ is the number of terms in the series.

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

A: To calculate the sum of a geometric series, you can use the formula for a finite geometric series:

S_n = \frac{a(1-r^n)}{1-r}

where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: What are some applications of geometric series?

A: Geometric series have numerous applications in mathematics and other fields, including:

Finance: Geometric series are used to calculate the future value of an investment.
Engineering: Geometric series are used to calculate the stress and strain on a material.
Computer Science: Geometric series are used to calculate the time complexity of an algorithm.

Q: Can I use geometric series to calculate the sum of an infinite series?

A: Yes, you can use geometric series to calculate the sum of an infinite series. However, the series must converge, meaning that the terms must approach zero as the number of terms increases.

Q: How do I determine if a geometric series converges?

A: To determine if a geometric series converges, you can use the ratio test. The ratio test states that if the absolute value of the common ratio $r$ is less than 1, the series converges.

Q: What is the ratio test?

A: The ratio test is a method used to determine if a series converges. The ratio test states that if the absolute value of the common ratio $r$ is less than 1, the series converges.

Q: Can I use geometric series to calculate the sum of a series with a variable common ratio?

A: No, you cannot use geometric series to calculate the sum of a series with a variable common ratio. Geometric series require a fixed common ratio to calculate the sum.

Q: How do I calculate the sum of a series with a variable common ratio?

A: To calculate the sum of a series with a variable common ratio, you can use other methods, such as the integral test or the comparison test.

Conclusion

S_n = \frac{a(1-r^n)}{1-r}

where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.