What Is The Sum Of The Geometric Series?$\sum_{n=1}^4(-2)(-3)^{n-1}$A. -122 B. -2 C. 40 D. 54

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 sum of a geometric series can be calculated using a formula, but it's also possible to calculate it manually by adding up the terms.

The Formula for the Sum of a Geometric Series

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
• $n$ is the number of terms

Calculating the Sum of a Geometric Series

To calculate the sum of a geometric series, we need to know the first term, the common ratio, and the number of terms. In this case, we are given the series:

$$\sum_{n=1}^4(-2)(-3)^{n-1}$$

This means that the first term is $a = (-2)(-3)^{0} = -2$, the common ratio is $r = -3$, and the number of terms is $n = 4$.

Calculating the Sum

To calculate the sum, we can plug these values into the formula:

$$S_4 = \frac{(-2)(1-(-3)^4)}{1-(-3)}$$

Simplifying the expression, we get:

$$S_4 = \frac{(-2)(1-81)}{4}$$

$$S_4 = \frac{(-2)(-80)}{4}$$

$$S_4 = 40$$

Conclusion

Therefore, the sum of the geometric series is 40.

Example Use Cases

Geometric series have many real-world applications, including:

• Finance: Geometric series can be used to calculate the future value of an investment or loan.
• Physics: Geometric series can be used to calculate the trajectory of a projectile or the motion of a pendulum.
• Computer Science: Geometric series can be used to calculate the time complexity of algorithms.

Common Mistakes

When calculating the sum of a geometric series, it's easy to make mistakes. Here are some common mistakes to avoid:

• Incorrectly identifying the first term: Make sure to identify the first term correctly, including any negative signs or exponents.
• Incorrectly identifying the common ratio: Make sure to identify the common ratio correctly, including any negative signs or exponents.
• Incorrectly applying the formula: Make sure to apply the formula correctly, including any simplifications or cancellations.

Conclusion

In conclusion, the sum of a geometric series can be calculated using a formula, but it's also possible to calculate it manually by adding up the terms. By understanding the formula and applying it correctly, we can calculate the sum of a geometric series with ease.

Final Answer

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Frequently Asked Questions

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

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

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

where:

• $S_n$ is the sum of the first $n$ terms
• $a$ is the first term
• $r$ is the common ratio
• $n$ is the number of terms

Q: What is the first term of a geometric series?

A: The first term of a geometric series is the first number in the sequence. It is denoted by $a$.

Q: What is the common ratio of a geometric series?

A: The common ratio of a geometric series is the number by which each term is multiplied to get the next term. It is denoted by $r$.

Q: How do I identify the common ratio of a geometric series?

A: To identify the common ratio of a geometric series, you need to look at the ratio of each term to the previous term. For example, if the series is:

$$2, 6, 18, 54, ...$$

The common ratio is $3$, because each term is multiplied by $3$ to get the next term.

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: What is the formula for the sum of an infinite geometric series?

A: The formula for the sum of an infinite geometric series is:

$$S = \frac{a}{1-r}$$

Q: When can I use the formula for the sum of an infinite geometric series?

A: You can use the formula for the sum of an infinite geometric series when the absolute value of the common ratio is less than $1$. This means that the series converges to a finite sum.

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 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 an arithmetic series?

A: To calculate the sum of an arithmetic series, you can use the formula:

$$S_n = \frac{n}{2}(a + l)$$

where:

• $S_n$ is the sum of the first $n$ terms
• $n$ is the number of terms
• $a$ is the first term
• $l$ is the last term

Q: What is the formula for the sum of a geometric series with a common ratio of $-1$?

A: The formula for the sum of a geometric series with a common ratio of $-1$ is:

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

Q: What is the formula for the sum of a geometric series with a common ratio of $1$?

A: The formula for the sum of a geometric series with a common ratio of $1$ is:

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

This formula is undefined, because the denominator is zero.

Conclusion

In conclusion, geometric series are a type of sequence 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, but it's also possible to calculate it manually by adding up the terms. By understanding the formula and applying it correctly, we can calculate the sum of a geometric series with ease.

Final Answer

The final answer is: $\boxed{40}$