Evaluate ∑ N = 1 5 3 ( − 2 ) N − 1 \sum_{n=1}^5 3(-2)^{n-1} ∑ N = 1 5 ​ 3 ( − 2 ) N − 1 .A. { -93$}$B. { -33$}$C. ${ 33\$} D. ${ 93\$}

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Introduction

In this article, we will evaluate the given summation $\sum_{n=1}^5 3(-2)^{n-1}$. This involves understanding the concept of summation, geometric series, and applying the formula for the sum of a finite geometric series. We will break down the problem step by step, and by the end of this article, we will have the final answer to the given summation.

Understanding the Summation

The given summation is $\sum_{n=1}^5 3(-2)^{n-1}$. This means we need to find the sum of the terms $3(-2)^{n-1}$ for $n$ ranging from $1$ to $5$. In other words, we need to find the sum of the following terms:

$3(-2)^{0}$, $3(-2)^{1}$, $3(-2)^{2}$, $3(-2)^{3}$, $3(-2)^{4}$

Evaluating the Terms

Let's evaluate each term individually:

• $3(-2)^{0} = 3(1) = 3$
• $3(-2)^{1} = 3(-2) = -6$
• $3(-2)^{2} = 3(4) = 12$
• $3(-2)^{3} = 3(-8) = -24$
• $3(-2)^{4} = 3(16) = 48$

Finding the Sum

Now that we have evaluated each term, we can find the sum by adding them together:

$3 + (-6) + 12 + (-24) + 48 = 33$

Conclusion

Therefore, the value of the given summation $\sum_{n=1}^5 3(-2)^{n-1}$ is $\boxed{33}$.

Final Answer

The final answer to the given summation is $\boxed{33}$.

Discussion

The given summation is a finite geometric series with a common ratio of $-2$. The formula for the sum of a finite geometric series is:

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

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

In this case, $a = 3$, $r = -2$, and $n = 5$. Plugging these values into the formula, we get:

$S_5 = \frac{3(1-(-2)^5)}{1-(-2)} = \frac{3(1-(-32))}{1+2} = \frac{3(33)}{3} = 33$

Therefore, the value of the given summation is $\boxed{33}$.

Related Topics

• Geometric series
• Finite geometric series
• Summation
• Algebra

References

• [1] "Geometric Series" by Math Is Fun
• [2] "Finite Geometric Series" by Khan Academy
• [3] "Summation" by Wolfram MathWorld
  

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Introduction

In our previous article, we evaluated the given summation $\sum_{n=1}^5 3(-2)^{n-1}$. In this article, we will answer some frequently asked questions related to the evaluation of this summation.

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 $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I apply the formula for the sum of a finite geometric series?

A: To apply the formula, you need to identify the first term $a$, the common ratio $r$, and the number of terms $n$. Then, plug these values into the formula and simplify.

Q: What is the first term $a$ in the given summation?

A: The first term $a$ is $3(-2)^{0}$, which is equal to $3$.

Q: What is the common ratio $r$ in the given summation?

A: The common ratio $r$ is $-2$.

Q: How many terms are there in the given summation?

A: There are $5$ terms in the given summation.

Q: Can I use the formula for the sum of a finite geometric series to evaluate the given summation?

A: Yes, you can use the formula to evaluate the given summation. The formula is:

$S_5 = \frac{3(1-(-2)^5)}{1-(-2)} = \frac{3(1-(-32))}{1+2} = \frac{3(33)}{3} = 33$

Q: What is the value of the given summation?

A: The value of the given summation is $\boxed{33}$.

Q: Can I use a calculator to evaluate the given summation?

A: Yes, you can use a calculator to evaluate the given summation. However, it's always a good idea to understand the underlying math and apply the formula manually.

Q: What are some common mistakes to avoid when evaluating a finite geometric series?

A: Some common mistakes to avoid when evaluating a finite geometric series include:

• Not identifying the first term $a$ correctly
• Not identifying the common ratio $r$ correctly
• Not identifying the number of terms $n$ correctly
• Not applying the formula correctly
• Not simplifying the expression correctly

Conclusion

In this article, we answered some frequently asked questions related to the evaluation of the given summation $\sum_{n=1}^5 3(-2)^{n-1}$. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer to the given summation is $\boxed{33}$.

Related Topics

• Geometric series
• Finite geometric series
• Summation
• Algebra

References

• [1] "Geometric Series" by Math Is Fun
• [2] "Finite Geometric Series" by Khan Academy
• [3] "Summation" by Wolfram MathWorld