Partial Sums of Geometric Series

Oct 13, 2025 by miodeya.com 68 views

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The partial sum of a geometric series is the sum of the first n terms of the series. In this article, we will explore which equation represents the partial sum of a geometric series.

What is a Geometric Series?

A geometric series is a sequence of numbers that can be written in the form:

a, ar, ar^2, ar^3, \ldots

where a is the first term and r is the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric series with a = 2 and r = 3.

Partial Sums of Geometric Series

The partial sum of a geometric series is the sum of the first n terms of the series. It can be calculated using the formula:

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

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

Analyzing the Options

Let's analyze the options given:

A. $\sum_{n=1}^4(125)\left(\frac{1}{5}\right)^{n-1}$

This option represents a geometric series with a = 125 and r = 1/5. The sum of the first 4 terms of this series can be calculated using the formula for partial sums:

S_4 = 125 \left(\frac{1-(1/5)^4}{1-1/5}\right)

Simplifying the expression, we get:

S_4 = 125 \left(\frac{1-1/625}{4/5}\right)

S_4 = 125 \left(\frac{624/625}{4/5}\right)

S_4 = 125 \left(\frac{624}{625} \times \frac{5}{4}\right)

S_4 = 125 \left(\frac{624}{500}\right)

S_4 = 125 \left(\frac{156}{125}\right)

S_4 = 156

B. $125+25+5+1$

This option represents the sum of the first 4 terms of a geometric series with a = 125 and r = 1/5. However, it does not represent the partial sum of the series, but rather the sum of the first 4 terms.

C. $25+5+1+\frac{1}{5}$

This option represents the sum of the first 4 terms of a geometric series with a = 25 and r = 1/5. However, it does not represent the partial sum of the series, but rather the sum of the first 4 terms.

D. $1+\frac{1}{5}+\frac{1}{25}+\frac{1}{125}$

This option represents the sum of the first 4 terms of a geometric series with a = 1 and r = 1/5. However, it does not represent the partial sum of the series, but rather the sum of the first 4 terms.

This option is blank and does not represent any geometric series.

Conclusion

Based on the analysis, option A represents the partial sum of a geometric series. The partial sum of the series is 156.

Understanding Geometric Series and Partial Sums: Key Takeaways

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
The partial sum of a geometric series is the sum of the first n terms of the series.
The partial sum of a geometric series can be calculated using the formula: $S_n = a \left(\frac{1-r^n}{1-r}\right)$
Option A represents the partial sum of a geometric series.

Frequently Asked Questions

What is a geometric series? A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
What is the partial sum of a geometric series? The partial sum of a geometric series is the sum of the first n terms of the series.
How is the partial sum of a geometric series calculated? The partial sum of a geometric series can be calculated using the formula: $S_n = a \left(\frac{1-r^n}{1-r}\right)$

Additional Resources

For more information on geometric series and partial sums, please refer to the following resources:

Khan Academy: Geometric Series
Math Is Fun: Geometric Series
Wolfram MathWorld: Geometric Series
Geometric Series and Partial Sums: Q&A =====================================

In this article, we will answer some frequently asked questions about geometric series and partial sums.

Q: What is a geometric series?

A: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the partial sum of a geometric series?

A: The partial sum of a geometric series is the sum of the first n terms of the series.

Q: How is the partial sum of a geometric series calculated?

A: The partial sum of a geometric series can be calculated using the formula: $S_n = a \left(\frac{1-r^n}{1-r}\right)$

Q: What is the formula for the partial sum of a geometric series?

A: The formula for the partial sum of a geometric series is: $S_n = a \left(\frac{1-r^n}{1-r}\right)$

Q: What is the common ratio in a geometric series?

A: The common ratio in a geometric series is the fixed, non-zero number that is multiplied by each term to get the next term.

Q: How do I determine the common ratio in a geometric series?

A: To determine the common ratio in a geometric series, you can divide any term by the previous term.

Q: What is the first term in a geometric series?

A: The first term in a geometric series is the first number in the sequence.

Q: How do I determine the first term in a geometric series?

A: To determine the first term in a geometric series, you can look at the first number in the sequence.

Q: What is the sum of an infinite geometric series?

A: The sum of an infinite geometric series is given by the formula: $S = \frac{a}{1-r}$

Q: What is the condition for the sum of an infinite geometric series to converge?

A: The sum of an infinite geometric series converges if and only if the absolute value of the common ratio is less than 1.

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

A: The formula for the sum of an infinite geometric series is: $S = \frac{a}{1-r}$

Q: How do I determine if the sum of an infinite geometric series converges?

A: To determine if the sum of an infinite geometric series converges, you can check if the absolute value of the common ratio is less than 1.

Q: What is the relationship between the sum of an infinite geometric series and the partial sum?

A: The sum of an infinite geometric series is equal to the limit of the partial sum as n approaches infinity.

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 formula: $S = \frac{a}{1-r}$

Q: What is the significance of the partial sum in a geometric series?

A: The partial sum in a geometric series represents the sum of the first n terms of the series.

Q: How do I use the partial sum in a geometric series?

A: You can use the partial sum in a geometric series to approximate the sum of the infinite series.

Q: What are some real-world applications of geometric series and partial sums?

A: Geometric series and partial sums have many real-world applications, including finance, economics, and engineering.

Conclusion

In this article, we have answered some frequently asked questions about geometric series and partial sums. We hope that this article has provided you with a better understanding of these concepts and their applications.

Additional Resources

For more information on geometric series and partial sums, please refer to the following resources:

Khan Academy: Geometric Series
Math Is Fun: Geometric Series
Wolfram MathWorld: Geometric Series

Frequently Asked Questions

What is a geometric series? A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
What is the partial sum of a geometric series? The partial sum of a geometric series is the sum of the first n terms of the series.
How is the partial sum of a geometric series calculated? The partial sum of a geometric series can be calculated using the formula: $S_n = a \left(\frac{1-r^n}{1-r}\right)$