Evaluate The Following:Find The First Term Of The Series:$\[ \sum_{n=1}^9 7\left(\frac{8}{5}\right)^{n-1} \\]The Formula For The Sum Of A Geometric Series Is Given By:$\[ S = \frac{a(1-r^n)}{1-r} \\]where \[$a\$\] Is The First

Introduction

In mathematics, 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 sum of a geometric series can be calculated using a specific formula, which is essential in various mathematical and real-world applications. In this article, we will evaluate the first term of a given geometric series using the formula for the sum of a geometric series.

The Formula for the Sum of a Geometric Series

The formula for the sum of a geometric series is given by:

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

where:

• $S$ is the sum of the series
• $a$ is the first term of the series
• $r$ is the common ratio
• $n$ is the number of terms in the series

Evaluating the First Term of the Series

The given series is:

$$\sum_{n=1}^9 7\left(\frac{8}{5}\right)^{n-1}$$

To evaluate the first term of this series, we need to use the formula for the sum of a geometric series. However, we are not given the sum of the series, but rather the formula for the sum of a geometric series. Therefore, we will use the formula to find the sum of the series and then use the result to find the first term.

Step 1: Find the Sum of the Series

Using the formula for the sum of a geometric series, we can find the sum of the given series as follows:

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

In this case, $a = 7$, $r = \frac{8}{5}$, and $n = 9$. Plugging these values into the formula, we get:

$$S = \frac{7\left(1-\left(\frac{8}{5}\right)^9\right)}{1-\frac{8}{5}}$$

Simplifying the expression, we get:

$$S = \frac{7\left(1-\left(\frac{8}{5}\right)^9\right)}{-\frac{3}{5}}$$

$$S = -\frac{35}{3}\left(1-\left(\frac{8}{5}\right)^9\right)$$

Step 2: Find the First Term of the Series

Now that we have found the sum of the series, we can use the formula for the sum of a geometric series to find the first term. The formula is:

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

Rearranging the formula to solve for $a$, we get:

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

Plugging in the values we found earlier, we get:

$$a = \frac{-\frac{35}{3}\left(1-\left(\frac{8}{5}\right)^9\right)\left(-\frac{3}{5}\right)}{1-\left(\frac{8}{5}\right)^9}$$

Simplifying the expression, we get:

$$a = 7$$

Therefore, the first term of the series is $7$.

Conclusion

In this article, we evaluated the first term of a given geometric series using the formula for the sum of a geometric series. We found that the sum of the series is $-\frac{35}{3}\left(1-\left(\frac{8}{5}\right)^9\right)$ and the first term of the series is $7$. This demonstrates the importance of the formula for the sum of a geometric series in evaluating the first term of a series.

References

• "Geometric Series" by Math Open Reference
• "Sum of a Geometric Series" by Wolfram MathWorld

Further Reading

• "Geometric Series and Sequences" by Khan Academy
• "Sum of a Geometric Series" by MIT OpenCourseWare
  Evaluating the First Term of a Geometric Series: Q&A =====================================================

Introduction

In our previous article, we evaluated the first term of a given geometric series using the formula for the sum of a geometric series. In this article, we will answer some frequently asked questions related to geometric series and the formula for the sum of a geometric series.

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 formula for the sum of a geometric series?

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

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

where:

• $S$ is the sum of the series
• $a$ is the first term of the series
• $r$ is the common ratio
• $n$ is the number of terms in the series

Q: How do I find the first term of a geometric series?

A: To find the first term of a geometric series, you can use the formula for the sum of a geometric series and rearrange it to solve for $a$. The formula is:

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

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 find the common ratio of a geometric series?

A: To find the common ratio of a geometric series, you can look at the ratio of any two consecutive terms. For example, if the series is $2, 6, 18, ...$, the common ratio is $\frac{6}{2} = 3$.

Q: What is the difference between a geometric series and an arithmetic 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. An arithmetic series, on the other hand, is a sequence of numbers where each term after the first is found by adding a fixed number called the common difference.

Q: How do I use the formula for the sum of a geometric series in real-world applications?

A: The formula for the sum of a geometric series has many real-world applications, such as calculating the future value of an investment, the amount of money that will be in a savings account after a certain number of years, or the amount of money that will be paid out in a series of payments.

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

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

• Not checking if the common ratio is positive or negative
• Not checking if the common ratio is greater than 1 or less than 1
• Not checking if the number of terms is a positive integer
• Not using the correct formula for the sum of a geometric series

Conclusion

In this article, we answered some frequently asked questions related to geometric series and the formula for the sum of a geometric series. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of geometric series.

References

• "Geometric Series" by Math Open Reference
• "Sum of a Geometric Series" by Wolfram MathWorld

Further Reading

• "Geometric Series and Sequences" by Khan Academy
• "Sum of a Geometric Series" by MIT OpenCourseWare