Sum of a Finite Geometric Series

Oct 13, 2025 by miodeya.com 34 views

When dealing with a series of numbers, it's essential to identify the pattern and understand how each term is generated. In this case, we have a series that starts with $3$ and then alternates between positive and negative terms. The common ratio between consecutive terms is $-3$ , as each term is obtained by multiplying the previous term by $-3$ .

Identifying the Pattern

To calculate the sum of the first five terms, we need to identify the pattern and understand how each term is generated. The series starts with $3$ , and then each subsequent term is obtained by multiplying the previous term by $-3$ . Therefore, the series can be written as:

3, -9, 27, -81, \cdots

Calculating the Sum

To calculate the sum of the first five terms, we can use the formula for the sum of a finite geometric series:

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

S_5 = \frac{3(1 - (-3)^5)}{1 - (-3)}

Simplifying the expression, we get:

S_5 = \frac{3(1 - (-243))}{4}

S_5 = \frac{3(244)}{4}

S_5 = 183

Conclusion

Therefore, the sum of the first five terms in the series is $183$ .

Answer

The correct answer is:

C. 183

Explanation

The series is a geometric series with a first term of $3$ and a common ratio of $-3$ . The sum of the first five terms can be calculated using the formula for the sum of a finite geometric series. By plugging in the values of $a$ , $r$ , and $n$ , we can calculate the sum of the first five terms, which is $183$ .

Additional Information

Geometric Series: A geometric series is a series 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.
Finite Geometric Series: A finite geometric series is a geometric series with a finite number of terms.
Sum of a Finite Geometric Series: The sum of a finite geometric series can be calculated using the formula $S_n = \frac{a(1 - r^n)}{1 - r}$ , where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
Q&A: Understanding Geometric Series and Calculating Sums =====================================================

Q: What is a geometric series?

A: A geometric series is a series 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 a finite geometric series?

A: A finite geometric series is a geometric series with a finite number of terms. In other words, it is a geometric series that has a specific number of terms, rather than an infinite number of terms.

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

A: The sum of a finite geometric series can be calculated using the formula $S_n = \frac{a(1 - r^n)}{1 - r}$ , where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: 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 significance of the common ratio (r) in a geometric series?

A: The common ratio (r) is a fixed, non-zero number that is used to generate each term in the series. It determines the rate at which the terms increase or decrease.

Q: How do I determine the common ratio (r) in a geometric series?

A: To determine the common ratio (r) in a geometric series, you can divide any term by its previous term. For example, if the series is 2, 6, 18, 54, ..., you can divide the second term (6) by the first term (2) to get the common ratio (r = 3).

Q: What is the significance of the first term (a) in a geometric series?

A: The first term (a) is the initial term in the series. It is the starting point for the series, and all subsequent terms are generated by multiplying the first term by the common ratio (r).

Q: How do I determine the first term (a) in a geometric series?

A: To determine the first term (a) in a geometric series, you can simply look at the first term in the series. For example, if the series is 3, -9, 27, -81, ..., the first term (a) is 3.

Q: What is the significance of the number of terms (n) in a geometric series?

A: The number of terms (n) is the total number of terms in the series. It determines how many terms are included in the sum.

Q: How do I determine the number of terms (n) in a geometric series?

A: To determine the number of terms (n) in a geometric series, you can simply count the number of terms in the series. For example, if the series is 2, 6, 18, 54, ..., the number of terms (n) is 4.

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

A: A geometric series is a series 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 series 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: The sum of an arithmetic series can be calculated using the formula $S_n = \frac{n(a + l)}{2}$ , where $n$ is the number of terms, $a$ is the first term, and $l$ is the last term.

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is:

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

Q: What is the significance of the last term (l) in an arithmetic series?

A: The last term (l) is the final term in the series. It is the last value in the series, and it is used to calculate the sum.

Q: How do I determine the last term (l) in an arithmetic series?

A: To determine the last term (l) in an arithmetic series, you can simply look at the last term in the series. For example, if the series is 2, 5, 8, 11, ..., the last term (l) is 11.