Write The Series Represented By ∑ N = 0 5 1 3 ( − 2 ) N \sum_{n=0}^5 \frac{1}{3}(-2)^n ∑ N = 0 5 ​ 3 1 ​ ( − 2 ) N . ∑ N = 0 5 1 3 ( − 2 ) N = □ \sum_{n=0}^5 \frac{1}{3}(-2)^n = \square ∑ N = 0 5 ​ 3 1 ​ ( − 2 ) N = □

Introduction

In mathematics, a geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant. The general form of a geometric series is given by $\sum_{n=0}^{\infty} ar^n$, where $a$ is the first term and $r$ is the common ratio. In this article, we will focus on representing and calculating a specific geometric series represented by $\sum_{n=0}^5 \frac{1}{3}(-2)^n$.

Understanding the Series

The given series is a geometric series with the first term $a = \frac{1}{3}$ and the common ratio $r = -2$. The series is finite, meaning it has a specific number of terms, which in this case is 6. The series can be written as:

$\sum_{n=0}^5 \frac{1}{3}(-2)^n = \frac{1}{3} + \frac{1}{3}(-2)^1 + \frac{1}{3}(-2)^2 + \frac{1}{3}(-2)^3 + \frac{1}{3}(-2)^4 + \frac{1}{3}(-2)^5$

Calculating the Series

To calculate the series, 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, we have $a = \frac{1}{3}$, $r = -2$, and $n = 6$. Plugging these values into the formula, we get:

$S_6 = \frac{\frac{1}{3}(1-(-2)^6)}{1-(-2)}$

Simplifying the Expression

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

$(-2)^6 = 64$

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

$S_6 = \frac{\frac{1}{3}(1-64)}{1-(-2)}$

Simplifying further, we get:

$S_6 = \frac{\frac{1}{3}(-63)}{3}$

Final Calculation

Now, we can perform the final calculation:

$S_6 = \frac{-21}{3}$

$S_6 = -7$

Conclusion

In this article, we represented and calculated a specific geometric series represented by $\sum_{n=0}^5 \frac{1}{3}(-2)^n$. We used the formula for the sum of a finite geometric series to calculate the series and obtained a final result of $-7$.

Geometric Series Formula

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.

Common Ratio and First Term

The common ratio $r$ is the constant by which each term is multiplied to obtain the next term. In this case, the common ratio is $-2$.

The first term $a$ is the first term of the series. In this case, the first term is $\frac{1}{3}$.

Geometric Series Representation

A geometric series can be represented in the form $\sum_{n=0}^{\infty} ar^n$, where $a$ is the first term and $r$ is the common ratio.

Geometric Series Formula Derivation

The formula for the sum of a finite geometric series can be derived by using the formula for the sum of a geometric series:

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

This formula can be used to calculate the sum of a finite geometric series.

Geometric Series Applications

Geometric series have many applications in mathematics and other fields. Some examples include:

• Finance: Geometric series can be used to calculate the future value of an investment.
• Physics: Geometric series can be used to calculate the motion of an object.
• Engineering: Geometric series can be used to calculate the stress and strain on a material.

Geometric Series Limitations

Geometric series have some limitations. For example:

• Convergence: Geometric series only converge if the absolute value of the common ratio is less than 1.
• Divergence: Geometric series diverge if the absolute value of the common ratio is greater than or equal to 1.

Conclusion

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Q: What is a geometric series?

A: A geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant. The general form of a geometric series is given by $\sum_{n=0}^{\infty} ar^n$, where $a$ is the first term and $r$ is 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: What is the common ratio?

A: The common ratio $r$ is the constant by which each term is multiplied to obtain the next term.

Q: What is the first term?

A: The first term $a$ is the first term of the series.

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

A: A geometric series is a series where each term is obtained by multiplying the previous term by a fixed constant, whereas an arithmetic series is a series where each term is obtained by adding a fixed constant to the previous term.

Q: When does a geometric series converge?

A: A geometric series converges if the absolute value of the common ratio is less than 1.

Q: When does a geometric series diverge?

A: A geometric series diverges if the absolute value of the common ratio is greater than or equal to 1.

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

A: To calculate the sum of a geometric series, you can use the formula for the sum of a finite geometric series:

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

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 calculate the future value of an investment.
• Physics: Geometric series can be used to calculate the motion of an object.
• Engineering: Geometric series can be used to calculate the stress and strain on a material.

Q: What are some limitations of geometric series?

A: Geometric series have some limitations, including:

• Convergence: Geometric series only converge if the absolute value of the common ratio is less than 1.
• Divergence: Geometric series diverge if the absolute value of the common ratio is greater than or equal to 1.

Q: Can I use geometric series to model real-world phenomena?

A: Yes, geometric series can be used to model real-world phenomena, such as population growth, compound interest, and the motion of objects.

Q: How do I determine if a geometric series is convergent or divergent?

A: To determine if a geometric series is convergent or divergent, you can use the following criteria:

• Convergence: If the absolute value of the common ratio is less than 1, the series converges.
• Divergence: If the absolute value of the common ratio is greater than or equal to 1, the series diverges.

Q: Can I use geometric series to solve problems in other fields?

A: Yes, geometric series can be used to solve problems in other fields, such as physics, engineering, and finance.

Conclusion

In this article, we answered some common questions about geometric series, including what a geometric series is, the formula for the sum of a finite geometric series, the common ratio, the first term, and the difference between a geometric series and an arithmetic series. We also discussed the convergence and divergence of geometric series, real-world applications, limitations, and how to determine if a geometric series is convergent or divergent.