Recall the Formula for the Sum of a Geometric Series

Oct 13, 2025 by miodeya.com 134 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 series is called geometric because the ratio of consecutive terms is constant. In this article, we will evaluate the given geometric series and find the value of S.

Understanding the Series

The given series is:

$\sum_{n=1}^{10} 5\left(\frac{1}{3}\right)^{n-1}$

This series has a common ratio of $\frac{1}{3}$ and the first term is $5$ . The series has 10 terms, and we need to find the sum of these terms.

Recall the Formula for the Sum of a Geometric Series

The formula for the sum of a geometric series is:

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

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

Apply the Formula to the Given Series

In this case, $a = 5$ , $r = \frac{1}{3}$ , and $n = 10$ . Plugging these values into the formula, we get:

$S = \frac{5(1-\left(\frac{1}{3}\right)^{10})}{1-\frac{1}{3}}$

Simplify the Expression

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

$\left(\frac{1}{3}\right)^{10} = \frac{1}{59049}$

Now, we can substitute this value back into the expression:

$S = \frac{5(1-\frac{1}{59049})}{1-\frac{1}{3}}$

Simplify the Expression Further

To simplify the expression further, we can start by evaluating the expression inside the parentheses:

$1-\frac{1}{59049} = \frac{59048}{59049}$

Now, we can substitute this value back into the expression:

$S = \frac{5\frac{59048}{59049}}{\frac{2}{3}}$

Simplify the Expression Even Further

To simplify the expression even further, we can start by multiplying the numerator and denominator by 3:

$S = \frac{5\frac{59048}{59049} \times 3}{2}$

Simplify the Expression to Get the Final Answer

To simplify the expression to get the final answer, we can start by multiplying the numerator and denominator:

$S = \frac{5 \times 59048 \times 3}{59049 \times 2}$

$S = \frac{885720}{118098}$

$S = \frac{885720}{118098} \times \frac{1}{1}$

$S = 7.5$

The final answer is $\boxed{7.5}$ .

Conclusion

In this article, we evaluated a geometric series and found the value of S. We used the formula for the sum of a geometric series and simplified the expression to get the final answer. The final answer is $\boxed{7.5}$ .

Introduction

Understanding the Series

The given series is:

$\sum_{n=1}^{10} 5\left(\frac{1}{3}\right)^{n-1}$

This series has a common ratio of $\frac{1}{3}$ and the first term is $5$ . The series has 10 terms, and we need to find the sum of these terms.

Recall the Formula for the Sum of a Geometric Series

The formula for the sum of a geometric series is:

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

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

Apply the Formula to the Given Series

In this case, $a = 5$ , $r = \frac{1}{3}$ , and $n = 10$ . Plugging these values into the formula, we get:

$S = \frac{5(1-\left(\frac{1}{3}\right)^{10})}{1-\frac{1}{3}}$

Simplify the Expression

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

$\left(\frac{1}{3}\right)^{10} = \frac{1}{59049}$

Now, we can substitute this value back into the expression:

$S = \frac{5(1-\frac{1}{59049})}{1-\frac{1}{3}}$

Simplify the Expression Further

To simplify the expression further, we can start by evaluating the expression inside the parentheses:

$1-\frac{1}{59049} = \frac{59048}{59049}$

Now, we can substitute this value back into the expression:

$S = \frac{5\frac{59048}{59049}}{\frac{2}{3}}$

Simplify the Expression Even Further

To simplify the expression even further, we can start by multiplying the numerator and denominator by 3:

$S = \frac{5\frac{59048}{59049} \times 3}{2}$

Simplify the Expression to Get the Final Answer

To simplify the expression to get the final answer, we can start by multiplying the numerator and denominator:

$S = \frac{5 \times 59048 \times 3}{59049 \times 2}$

$S = \frac{885720}{118098}$

$S = \frac{885720}{118098} \times \frac{1}{1}$

$S = 7.5$

The final answer is $\boxed{7.5}$ .

Conclusion

Q&A

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.

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

A: The formula for the sum of a geometric series is:

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

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

Q: How do I apply the formula to a given series?

A: To apply the formula, you need to plug in the values of $a$ , $r$ , and $n$ into the formula.

Q: What is the common ratio in the given series?

A: The common ratio in the given series is $\frac{1}{3}$ .

Q: How do I simplify the expression to get the final answer?

A: To simplify the expression, you can start by evaluating the exponent, then substitute the value back into the expression, and finally simplify the expression further by multiplying the numerator and denominator.

Q: What is the final answer to the given series?

A: The final answer to the given series is $\boxed{7.5}$ .

Q: Can I use the formula for the sum of a geometric series to evaluate any series?

A: Yes, you can use the formula for the sum of a geometric series to evaluate any series that is a geometric series.

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

A: The first term in a geometric series is the starting point of the series, and it is used to calculate the sum of the series.

Q: How do I determine the number of terms in a geometric series?

A: The number of terms in a geometric series is usually given in the problem statement.

Q: Can I use the formula for the sum of a geometric series to evaluate a series with a negative common ratio?

A: Yes, you can use the formula for the sum of a geometric series to evaluate a series with a negative common ratio.

Q: What is the difference between a geometric series and an arithmetic series?

A: A geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant, while an arithmetic series is a type of series where each term is obtained by adding a fixed constant to the previous term.