Finding the Correct Summation

Oct 13, 2025 by magicalceylontours.com 51 views

Understanding 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 general form of a geometric series is given by:

${a, ar, ar^2, ar^3, \ldots, ar^{n-1}}$

where $a$ is the first term and $r$ is the common ratio.

The Given Series

The given series is:

${200 + 100 + 50 + \cdots + \frac{25}{8}}$

This series can be written in a more compact form using sigma notation:

${\sum_{n=1}^5 200\left(\frac{1}{2}\right)^{n-1}}$

However, this is not the correct summation for the given series.

Correcting the Summation

To find the correct summation, we need to identify the first term and the common ratio of the series. In this case, the first term is 200 and the common ratio is $\frac{1}{2}$ .

The correct summation for the given series is:

${\sum_{n=1}^5 200\left(\frac{1}{2}\right)^{n-1}}$

However, this summation only goes up to $n=5$ , which means it only includes the first 5 terms of the series. To find the correct summation, we need to include all the terms of the series.

Finding the Correct Summation

To find the correct summation, we can use the formula for the sum of a finite geometric series:

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

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.

In this case, $a=200$ , $r=\frac{1}{2}$ , and $n=5$ . Plugging these values into the formula, we get:

${S_5 = \frac{200\left(1-\left(\frac{1}{2}\right)^5\right)}{1-\frac{1}{2}}}$

Simplifying this expression, we get:

${S_5 = \frac{200\left(1-\frac{1}{32}\right)}{\frac{1}{2}}}$

${S_5 = \frac{200\left(\frac{31}{32}\right)}{\frac{1}{2}}}$

${S_5 = 200\left(\frac{31}{32}\right)\left(2\right)}$

${S_5 = 200\left(\frac{31}{16}\right)}$

${S_5 = 200\left(1.9375\right)}$

${S_5 = 387.5}$

Conclusion

The correct summation for the given geometric series is:

${S_5 = 387.5}$

This is the sum of the first 5 terms of the series, and it is the correct answer to the problem.

Discussion

The problem of finding the correct summation for a geometric series is an important one in mathematics. It requires a good understanding of the formula for the sum of a finite geometric series and the ability to apply it to a given problem.

In this case, the student incorrectly wrote the summation for the given geometric series. However, by using the formula for the sum of a finite geometric series, we were able to find the correct summation.

This problem is a good example of how mathematics can be used to solve real-world problems. It requires a good understanding of mathematical concepts and the ability to apply them to a given problem.

Applications

The concept of geometric series has many applications in mathematics and other fields. Some examples include:

Finance: Geometric series can be used to calculate the future value of an investment or the present value of a future payment.
Engineering: Geometric series can be used to calculate the sum of a series of measurements or the sum of a series of forces.
Computer Science: Geometric series can be used to calculate the sum of a series of numbers or the sum of a series of operations.

Future Work

In the future, it would be interesting to explore other applications of geometric series. Some possible areas of research include:

Using geometric series to solve problems in physics: Geometric series can be used to calculate the sum of a series of measurements or the sum of a series of forces. This could be useful in solving problems in physics, such as calculating the trajectory of a projectile.
Using geometric series to solve problems in engineering: Geometric series can be used to calculate the sum of a series of measurements or the sum of a series of forces. This could be useful in solving problems in engineering, such as calculating the stress on a beam.
Using geometric series to solve problems in computer science: Geometric series can be used to calculate the sum of a series of numbers or the sum of a series of operations. This could be useful in solving problems in computer science, such as calculating the time complexity of an algorithm.

Conclusion

In conclusion, the problem of finding the correct summation for a geometric series is an important one in mathematics. It requires a good understanding of the formula for the sum of a finite geometric series and the ability to apply it to a given problem.

By using the formula for the sum of a finite geometric series, we were able to find the correct summation for the given series. This problem is a good example of how mathematics can be used to solve real-world problems.

The concept of geometric series has many applications in mathematics and other fields. Some examples include finance, engineering, and computer science.

In the future, it would be interesting to explore other applications of geometric series. Some possible areas of research include using geometric series to solve problems in physics, engineering, and computer science.

Understanding Geometric Series

${a, ar, ar^2, ar^3, \ldots, ar^{n-1}}$

where $a$ is the first term and $r$ is the common ratio.

Q&A

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:

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

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 correct summation for a geometric series?

A: To find the correct summation for a geometric series, you need to identify the first term and the common ratio of the series. Then, you can use the formula for the sum of a finite geometric series to find the correct summation.

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

Q: How do I calculate the sum of a geometric series using a calculator?

A: To calculate the sum of a geometric series using a calculator, you can use the formula for the sum of a finite geometric series. Simply plug in the values of the first term, the common ratio, and the number of terms into the formula, and the calculator will give you the sum.

Q: Can I use a geometric series to solve problems in finance?

A: Yes, you can use a geometric series to solve problems in finance. For example, you can use a geometric series to calculate the future value of an investment or the present value of a future payment.

Q: Can I use a geometric series to solve problems in engineering?

A: Yes, you can use a geometric series to solve problems in engineering. For example, you can use a geometric series to calculate the sum of a series of measurements or the sum of a series of forces.

Q: Can I use a geometric series to solve problems in computer science?

A: Yes, you can use a geometric series to solve problems in computer science. For example, you can use a geometric series to calculate the sum of a series of numbers or the sum of a series of operations.

Conclusion

In conclusion, geometric series are an important concept in mathematics that have many applications in finance, engineering, and computer science. By understanding the formula for the sum of a finite geometric series and how to apply it to a given problem, you can use geometric series to solve a wide range of problems.

Frequently Asked Questions

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:

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

Q: How do I find the correct summation for a geometric series?

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

Additional Resources

Geometric Series Formula: The formula for the sum of a finite geometric series is:

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

Geometric Series Calculator: You can use a calculator to calculate the sum of a geometric series by plugging in the values of the first term, the common ratio, and the number of terms into the formula.
Geometric Series Examples: You can find examples of geometric series in finance, engineering, and computer science by searching online or in textbooks.