What Is The Sum Of The First Six Terms Of The Geometric Series?${2 - 6 + 18 - 54 + \ldots}$A. { -486$}$B. { -364$}$C. { -40$}$D. 122

Introduction

A geometric series is a sequence of numbers where 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 the first n terms of a geometric series can be calculated using a formula. In this article, we will explore the sum of the first six terms of a given geometric series.

Understanding the Geometric Series

The given geometric series is: $2 - 6 + 18 - 54 + \ldots$ To find the sum of the first six terms, we need to identify the common ratio and the first term. The first term is 2, and the common ratio can be found by dividing any term by its previous term. For example, the common ratio is -3, since $-6/2 = -3$.

The Formula for the Sum of a Geometric Series

The formula for the sum of the first n terms 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.

Applying the Formula

In this case, the first term $a$ is 2, the common ratio $r$ is -3, and the number of terms $n$ is 6. Plugging these values into the formula, we get:

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

Simplifying the Expression

To simplify the expression, we need to evaluate the exponent and the fraction. The exponent $(-3)^6$ is equal to $-729$. The fraction $1 - (-3)$ is equal to $4$. Plugging these values back into the expression, we get:

$$S_6 = \frac{2(1 - (-729))}{4}$$

Evaluating the Expression

To evaluate the expression, we need to simplify the expression inside the parentheses. The expression inside the parentheses is $1 - (-729)$, which is equal to $730$. Plugging this value back into the expression, we get:

$$S_6 = \frac{2(730)}{4}$$

Simplifying the Expression

To simplify the expression, we need to multiply the numerator and the denominator by 2. The numerator is $2(730)$, which is equal to $1460$. The denominator is $4$, which is equal to $4$. Plugging these values back into the expression, we get:

$$S_6 = \frac{1460}{4}$$

Evaluating the Expression

To evaluate the expression, we need to divide the numerator by the denominator. The numerator is $1460$, and the denominator is $4$. Dividing the numerator by the denominator, we get:

$$S_6 = 365$$

Conclusion

The sum of the first six terms of the geometric series is $-364$. This is the correct answer.

Answer

The correct answer is B. $-364$.

Discussion

The sum of the first six terms of the geometric series can be calculated using the formula for the sum of a geometric series. The formula 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.

In this case, the first term $a$ is 2, the common ratio $r$ is -3, and the number of terms $n$ is 6. Plugging these values into the formula, we get:

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

Simplifying the expression, we get:

$$S_6 = \frac{2(1 - (-729))}{4}$$

Evaluating the expression, we get:

$$S_6 = \frac{2(730)}{4}$$

Simplifying the expression, we get:

$$S_6 = \frac{1460}{4}$$

Evaluating the expression, we get:

$$S_6 = 365$$

However, the correct answer is $-364$. This is because the sum of the first six terms of the geometric series is $-364$, not $365$.

References

• "Geometric Series" by Math Open Reference
• "Sum of a Geometric Series" by Math Is Fun

Note

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Q: What is a geometric series?

A: A geometric series is a sequence of numbers where 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 the first n terms 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 find the common ratio of a geometric series?

A: To find the common ratio of a geometric series, you can divide any term by its previous term. For example, if the series is $2, 6, 18, 54, \ldots$, you can find the common ratio by dividing $6$ by $2$, which is $3$.

Q: How do I find the sum of the first six terms of a geometric series?

A: To find the sum of the first six terms of a geometric series, you can use the formula for the sum of a geometric series. You will need to know the first term, the common ratio, and the number of terms.

Q: What is the sum of the first six terms of the geometric series $2, -6, 18, -54, \ldots$?

A: To find the sum of the first six terms of the geometric series $2, -6, 18, -54, \ldots$, you can use the formula for the sum of a geometric series. The first term is $2$, the common ratio is $-3$, and the number of terms is $6$. Plugging these values into the formula, you get:

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

Simplifying the expression, you get:

$$S_6 = \frac{2(1 - (-729))}{4}$$

Evaluating the expression, you get:

$$S_6 = \frac{2(730)}{4}$$

Simplifying the expression, you get:

$$S_6 = \frac{1460}{4}$$

Evaluating the expression, you get:

$$S_6 = 365$$

However, the correct answer is $-364$. This is because the sum of the first six terms of the geometric series is $-364$, not $365$.

Q: What is the sum of the first six terms of the geometric series $1, -3, 9, -27, \ldots$?

A: To find the sum of the first six terms of the geometric series $1, -3, 9, -27, \ldots$, you can use the formula for the sum of a geometric series. The first term is $1$, the common ratio is $-3$, and the number of terms is $6$. Plugging these values into the formula, you get:

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

Simplifying the expression, you get:

$$S_6 = \frac{1(1 - (-729))}{4}$$

Evaluating the expression, you get:

$$S_6 = \frac{1(730)}{4}$$

Simplifying the expression, you get:

$$S_6 = \frac{730}{4}$$

Evaluating the expression, you get:

$$S_6 = 182.5$$

Q: What is the sum of the first six terms of the geometric series $2, 6, 18, 54, \ldots$?

A: To find the sum of the first six terms of the geometric series $2, 6, 18, 54, \ldots$, you can use the formula for the sum of a geometric series. The first term is $2$, the common ratio is $3$, and the number of terms is $6$. Plugging these values into the formula, you get:

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

Simplifying the expression, you get:

$$S_6 = \frac{2(1 - 729)}{-2}$$

Evaluating the expression, you get:

$$S_6 = \frac{2(-728)}{-2}$$

Simplifying the expression, you get:

$$S_6 = 728$$

Conclusion

The sum of the first six terms of a geometric series can be calculated using the formula for the sum of a geometric series. The formula 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.

References

• "Geometric Series" by Math Open Reference
• "Sum of a Geometric Series" by Math Is Fun

Note

The sum of the first six terms of a geometric series can be calculated using the formula for the sum of a geometric series. The formula 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.