Evaluate:$\[ \sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1} \\]$\[ S = [?] \\]

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Introduction

In mathematics, a series is a sequence of numbers that are added together. A finite series is a series that has a finite number of terms, and it can be evaluated using various methods. In this article, we will evaluate the finite series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ and find its sum.

Understanding the Series

The given series is a geometric series, which is a series of the form $\sum_{n=1}^{N} ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio. In this case, the first term $a$ is $8$ and the common ratio $r$ is $\frac{1}{4}$.

Formula for the Sum of a Geometric Series

The sum of a geometric series can be found using the formula:

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

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

Evaluating the Series

We can now use the formula to evaluate the given series. Plugging in the values, we get:

$$S = \frac{8\left(1-\left(\frac{1}{4}\right)^{10}\right)}{1-\frac{1}{4}}$$

Simplifying the Expression

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

$$\left(\frac{1}{4}\right)^{10} = \frac{1}{1048576}$$

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

$$S = \frac{8\left(1-\frac{1}{1048576}\right)}{\frac{3}{4}}$$

Further Simplification

To further simplify the expression, we can multiply the numerator and denominator by $4$:

$$S = \frac{32\left(1-\frac{1}{1048576}\right)}{3}$$

Evaluating the Expression

Now, we can evaluate the expression:

$$S = \frac{32\left(\frac{1048575}{1048576}\right)}{3}$$

Simplifying the Fraction

To simplify the fraction, we can multiply the numerator and denominator by $1048576$:

$$S = \frac{32\left(\frac{1048575}{1048576}\right)}{3} = \frac{32\left(\frac{1048575}{1048576}\right)}{3}$$

Final Evaluation

To get the final answer, we can multiply the numerator and denominator by $3$:

$$S = \frac{32\left(\frac{1048575}{1048576}\right)}{3} = \frac{32\left(\frac{1048575}{1048576}\right)}{3}$$

Conclusion

In this article, we evaluated the finite series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ and found its sum. The sum of the series is $\frac{32\left(\frac{1048575}{1048576}\right)}{3}$. This result can be used in various mathematical applications, such as in the calculation of interest rates and in the analysis of financial data.

Discussion

The evaluation of the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ is an important problem in mathematics, as it can be used to model real-world situations. For example, the series can be used to calculate the amount of money that will be accumulated in a savings account after a certain number of years, assuming a fixed interest rate.

Applications

The series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ has many applications in mathematics and finance. Some of the applications include:

• Interest Rates: The series can be used to calculate the interest rate on a savings account or a loan.
• Financial Analysis: The series can be used to analyze financial data and make predictions about future trends.
• Probability: The series can be used to calculate probabilities in probability theory.

Future Work

In the future, we can use the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ to model more complex real-world situations. For example, we can use the series to calculate the amount of money that will be accumulated in a savings account after a certain number of years, assuming a variable interest rate.

References

• Mathematics: The series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ is a classic problem in mathematics, and it has been studied by many mathematicians throughout history.
• Finance: The series can be used to calculate interest rates and analyze financial data.

Conclusion

In conclusion, the evaluation of the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ is an important problem in mathematics, and it has many applications in finance and probability theory. The sum of the series is $\frac{32\left(\frac{1048575}{1048576}\right)}{3}$, and it can be used to model real-world situations and make predictions about future trends.

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Introduction

In our previous article, we evaluated the finite series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ and found its sum. In this article, we will answer some frequently asked questions about the series and its evaluation.

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 $S$ is the sum, $a$ is the first term, $r$ is the common ratio, and $N$ is the number of terms.

Q: How do I evaluate the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$?

A: To evaluate the series, you can use the formula for the sum of a geometric series. Plug in the values of $a$, $r$, and $N$ into the formula, and simplify the expression.

Q: What is the sum of the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$?

A: The sum of the series is $\frac{32\left(\frac{1048575}{1048576}\right)}{3}$.

Q: Can I use the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ to model real-world situations?

A: Yes, the series can be used to model real-world situations, such as calculating interest rates and analyzing financial data.

Q: What are some applications of the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$?

A: Some applications of the series include:

• Interest Rates: The series can be used to calculate the interest rate on a savings account or a loan.
• Financial Analysis: The series can be used to analyze financial data and make predictions about future trends.
• Probability: The series can be used to calculate probabilities in probability theory.

Q: Can I use the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ to calculate the amount of money that will be accumulated in a savings account after a certain number of years?

A: Yes, the series can be used to calculate the amount of money that will be accumulated in a savings account after a certain number of years, assuming a fixed interest rate.

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

A: A geometric series is a series of the form $\sum_{n=1}^{N} ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio. An arithmetic series is a series of the form $\sum_{n=1}^{N} a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.

Q: Can I use the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ to calculate the amount of money that will be accumulated in a savings account after a certain number of years, assuming a variable interest rate?

A: No, the series is only valid for a fixed interest rate. To calculate the amount of money that will be accumulated in a savings account after a certain number of years, assuming a variable interest rate, you would need to use a different formula.

Conclusion

In this article, we answered some frequently asked questions about the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ and its evaluation. We hope that this article has been helpful in understanding the series and its applications.

Discussion

The series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ is a classic problem in mathematics, and it has many applications in finance and probability theory. The sum of the series is $\frac{32\left(\frac{1048575}{1048576}\right)}{3}$, and it can be used to model real-world situations and make predictions about future trends.

References

• Mathematics: The series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ is a classic problem in mathematics, and it has been studied by many mathematicians throughout history.
• Finance: The series can be used to calculate interest rates and analyze financial data.

Future Work

In the future, we can use the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ to model more complex real-world situations. For example, we can use the series to calculate the amount of money that will be accumulated in a savings account after a certain number of years, assuming a variable interest rate.

Conclusion

In conclusion, the series $\sum_{n=1}^{10} 8\left(\frac{1}{4}\right)^{n-1}$ is a classic problem in mathematics, and it has many applications in finance and probability theory. The sum of the series is $\frac{32\left(\frac{1048575}{1048576}\right)}{3}$, and it can be used to model real-world situations and make predictions about future trends.