Find The Sum, If It Exists: ∑ K = 1 12 17 ( 2 7 ) K \sum_{k=1}^{12} 17\left(\frac{2}{7}\right)^k ∑ K = 1 12 ​ 17 ( 7 2 ​ ) K Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The Sum Is Approximately □ \square □ (Type An Integer Or A

Introduction

In this article, we will explore the concept of finding the sum of a geometric series. A geometric series is a sequence of numbers in which 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 formula, but in this case, we will use the formula for the sum of a finite geometric series to find the sum of the given series.

The Formula for the Sum of a Finite Geometric Series

The formula for the sum of 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.

The Given Series

The given series is:

$$\sum_{k=1}^{12} 17\left(\frac{2}{7}\right)^k$$

This is a geometric series with a first term of $17\left(\frac{2}{7}\right)^1$ and a common ratio of $\frac{2}{7}$. We need to find the sum of the first 12 terms of this series.

Finding the Sum

To find the sum, we can plug in the values into the formula for the sum of a finite geometric series:

$$S_{12} = \frac{17\left(\frac{2}{7}\right)^1(1-\left(\frac{2}{7}\right)^{12})}{1-\frac{2}{7}}$$

Simplifying the expression, we get:

$$S_{12} = \frac{17\left(\frac{2}{7}\right)(1-\left(\frac{2}{7}\right)^{12})}{\frac{5}{7}}$$

$$S_{12} = \frac{17\left(\frac{2}{7}\right)(1-\left(\frac{2}{7}\right)^{12})}{\frac{5}{7}} \times \frac{7}{7}$$

$$S_{12} = \frac{17 \times 2 \times (1-\left(\frac{2}{7}\right)^{12})}{5}$$

$$S_{12} = \frac{34 \times (1-\left(\frac{2}{7}\right)^{12})}{5}$$

Now, we need to calculate the value of $\left(\frac{2}{7}\right)^{12}$.

Calculating the Value of $\left(\frac{2}{7}\right)^{12}$

To calculate the value of $\left(\frac{2}{7}\right)^{12}$, we can use a calculator or a computer program. The value of $\left(\frac{2}{7}\right)^{12}$ is approximately:

$$\left(\frac{2}{7}\right)^{12} \approx 0.0000064$$

Now, we can plug in this value into the expression for $S_{12}$:

$$S_{12} = \frac{34 \times (1-0.0000064)}{5}$$

$$S_{12} = \frac{34 \times 0.9999936}{5}$$

$$S_{12} = \frac{34 \times 0.9999936}{5} \times \frac{1}{1}$$

$$S_{12} = \frac{34 \times 0.9999936}{5}$$

$$S_{12} = 6.799$$

Conclusion

The sum of the given series is approximately $6.799$. This is the final answer.

Answer

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Q: What is a geometric series?

A: A geometric series is a sequence of numbers in which 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 finite geometric series?

A: The formula for the sum of 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: How do I find the sum of a geometric series?

A: To find the sum of a geometric series, you can use the formula for the sum of a finite geometric series. You will need to know the first term, the common ratio, and the number of terms.

Q: What is the common ratio?

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

Q: How do I calculate the common ratio?

A: To calculate the common ratio, you can divide any term by the previous term. For example, if the series is 2, 6, 18, 54, ..., the common ratio is 3, since each term is 3 times the previous term.

Q: What is the first term?

A: The first term is the first number in the series.

Q: How do I find the first term?

A: To find the first term, you can look at the series and identify the first number. For example, if the series is 2, 6, 18, 54, ..., the first term is 2.

Q: What is the number of terms?

A: The number of terms is the total number of terms in the series.

Q: How do I find the number of terms?

A: To find the number of terms, you can count the number of terms in the series. For example, if the series is 2, 6, 18, 54, ..., the number of terms is 4.

Q: Can I use a calculator to find the sum of a geometric series?

A: Yes, you can use a calculator to find the sum of a geometric series. Simply enter the values into the calculator and follow the instructions to find the sum.

Q: Can I use a computer program to find the sum of a geometric series?

A: Yes, you can use a computer program to find the sum of a geometric series. Simply enter the values into the program and follow the instructions to find the sum.

Q: What if the series has a negative common ratio?

A: If the series has a negative common ratio, the sum of the series will be negative. To find the sum, you can use the formula for the sum of a finite geometric series, but you will need to take the absolute value of the common ratio.

Q: What if the series has a common ratio of 1?

A: If the series has a common ratio of 1, the series is not geometric. To find the sum, you can use the formula for the sum of an arithmetic series.

Q: What if the series has a common ratio of -1?

A: If the series has a common ratio of -1, the series is alternating. To find the sum, you can use the formula for the sum of an alternating series.

Conclusion

In this article, we have discussed the concept of geometric series and how to find the sum of a geometric series. We have also answered some frequently asked questions about geometric series.