Evaluate The Sum: $\sum_{n=1}^6 200\left(\frac{3}{2}\right)^{n-1}$

Feb 26, 2025 by ADMIN 67 views

**Evaluating the Sum of a Geometric Series: A Step-by-Step Guide**

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Introduction

In mathematics, 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, which is a powerful tool for solving problems involving exponential growth or decay. In this article, we will evaluate the sum of a specific geometric series: $\sum_{n=1}^6 200\left(\frac{3}{2}\right)^{n-1}$ .

Understanding the Series

The given series is a geometric series with the first term $a = 200$ and the common ratio $r = \frac{3}{2}$ . The series has 6 terms, and we need to find the sum of these terms. To do this, 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.

Applying the Formula

Now, let's apply the formula to the given series. We have:

$a = 200$
$r = \frac{3}{2}$
$n = 6$

Substituting these values into the formula, we get:

S_6 = \frac{200\left(1 - \left(\frac{3}{2}\right)^6\right)}{1 - \frac{3}{2}}

Simplifying the Expression

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

\left(\frac{3}{2}\right)^6 = \frac{3^6}{2^6} = \frac{729}{64}

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

S_6 = \frac{200\left(1 - \frac{729}{64}\right)}{1 - \frac{3}{2}}

Continuing the Simplification

To continue simplifying the expression, we can start by evaluating the denominator:

1 - \frac{3}{2} = -\frac{1}{2}

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

S_6 = \frac{200\left(1 - \frac{729}{64}\right)}{-\frac{1}{2}}

Further Simplification

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

1 - \frac{729}{64} = \frac{64 - 729}{64} = -\frac{665}{64}

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

S_6 = \frac{200\left(-\frac{665}{64}\right)}{-\frac{1}{2}}

Final Simplification

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

S_6 = 200 \times \frac{665}{64} \times \frac{2}{1}

S_6 = 200 \times \frac{1330}{64}

S_6 = \frac{266000}{64}

S_6 = 4156.25

Conclusion

In this article, we evaluated the sum of a geometric series using the formula for the sum of a finite geometric series. We started by understanding the series and identifying the first term, common ratio, and number of terms. We then applied the formula to the series and simplified the expression step by step. Finally, we obtained the final answer, which is the sum of the series.

Key Takeaways

The sum of a geometric series can be calculated using the formula: $S_n = \frac{a(1 - r^n)}{1 - r}$
The formula requires the first term, common ratio, and number of terms as input.
The series can be simplified by evaluating the exponent and denominator.
The final answer can be obtained by multiplying the numerator and denominator.

Real-World Applications

Geometric series have many real-world applications, including:

Finance: Geometric series can be used to model the growth of investments or the decay of debts.
Biology: Geometric series can be used to model the growth of populations or the decay of radioactive materials.
Engineering: Geometric series can be used to model the behavior of electrical circuits or the growth of chemical reactions.

Future Directions

In the future, we can explore more advanced topics in geometric series, such as:

Infinite geometric series: We can study the sum of an infinite geometric series and explore the conditions under which the series converges.
Geometric series with complex numbers: We can study the sum of geometric series with complex numbers and explore the properties of these series.
Applications of geometric series: We can explore more real-world applications of geometric series and develop new models and techniques for solving problems.

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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 geometric series?

A: The formula for the sum of a 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 determine the first term, common ratio, and number of terms in a geometric series?

A: To determine the first term, common ratio, and number of terms in a geometric series, you need to examine the series and identify the pattern. The first term is the first number in the series, the common ratio is the number that is multiplied by the previous term to get the next term, and the number of terms is the total number of terms in the series.

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: Can a geometric series have a negative common ratio?

A: Yes, a geometric series can have a negative common ratio. In this case, the series will converge to a finite sum if the absolute value of the common ratio is less than 1.

Q: How do I determine if a geometric series converges or diverges?

A: To determine if a geometric series converges or diverges, you need to examine the common ratio. If the absolute value of the common ratio is less than 1, the series converges to a finite sum. If the absolute value of the common ratio is greater than or equal to 1, the series diverges.

Q: What are some real-world applications of geometric series?

A: Geometric series have many real-world applications, including:

Finance: Geometric series can be used to model the growth of investments or the decay of debts.
Biology: Geometric series can be used to model the growth of populations or the decay of radioactive materials.
Engineering: Geometric series can be used to model the behavior of electrical circuits or the growth of chemical reactions.

Q: How do I use a geometric series to model real-world problems?

A: To use a geometric series to model real-world problems, you need to identify the pattern of growth or decay in the problem and determine the first term, common ratio, and number of terms. You can then use the formula for the sum of a geometric series to calculate the total amount or quantity.

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:

Incorrectly identifying the first term, common ratio, and number of terms
Failing to check for convergence or divergence
Using the wrong formula or method

Q: How do I troubleshoot common issues with geometric series?

A: To troubleshoot common issues with geometric series, you need to carefully examine the series and identify the problem. You can then use the formula for the sum of a geometric series or other methods to solve the problem.

Q: What are some advanced topics in geometric series?

A: Some advanced topics in geometric series include:

Infinite geometric series: We can study the sum of an infinite geometric series and explore the conditions under which the series converges.
Geometric series with complex numbers: We can study the sum of geometric series with complex numbers and explore the properties of these series.
Applications of geometric series: We can explore more real-world applications of geometric series and develop new models and techniques for solving problems.