Select The Correct Answer.What Is The Sum Of This Geometric Series? ∑ K = 1 4 6 K − 1 \sum_{k=1}^4 6^{k-1} ∑ K = 1 4 6 K − 1 A. 43 B. 1,554 C. 259 D. 1,295
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 a geometric series can be calculated using a formula, which is a crucial concept in mathematics, particularly in algebra and calculus. In this article, we will explore the sum of a geometric series and apply it to a specific problem.
What is a Geometric Series?
A geometric series is a sequence of numbers that can be written in the form:
a, ar, ar^2, ar^3, ...
where 'a' is the first term and 'r' is the common ratio. The sum of a geometric series can be calculated using the formula:
S = a + ar + ar^2 + ar^3 + ... + ar^(n-1)
where 'S' is the sum of the series, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
The Formula for the Sum of a Geometric Series
The formula for the sum of a geometric series is:
S = a * (1 - r^n) / (1 - r)
where 'S' is the sum of the series, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Applying the Formula to the Given Problem
The given problem is:
This is a geometric series with a first term of 1 (since 6^0 = 1) and a common ratio of 6. The number of terms is 4.
Using the formula for the sum of a geometric series, we can calculate the sum as follows:
S = 1 * (1 - 6^4) / (1 - 6)
S = (1 - 1296) / (-5)
S = -1295 / -5
S = 259
Conclusion
In conclusion, the sum of the given geometric series is 259. This is calculated using the formula for the sum of a geometric series, which is a crucial concept in mathematics. Understanding the sum of a geometric series is essential in various fields, including algebra, calculus, and engineering.
Answer
The correct answer is C. 259.
Additional Information
- Geometric series are used in various fields, including finance, economics, and engineering.
- The sum of a geometric series can be calculated using the formula: S = a * (1 - r^n) / (1 - r)
- The formula for the sum of a geometric series is a crucial concept in mathematics.
References
Q&A: Geometric Series
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 a geometric series is:
S = a * (1 - r^n) / (1 - r)
where 'S' is the sum of the series, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Q: How do I calculate the sum of a geometric series?
A: To calculate the sum of a geometric series, you need to know the first term, the common ratio, and the number of terms. You can use the formula above to calculate the sum.
Q: What is the common ratio in a geometric series?
A: The common ratio is the fixed, non-zero number that is multiplied by each term to get the next term in the series.
Q: How do I find the common ratio in a geometric series?
A: To find the common ratio, you can divide any term by the previous term. For example, if the series is 2, 6, 18, 54, ..., the common ratio is 3 (since 6/2 = 3, 18/6 = 3, and 54/18 = 3).
Q: What is the first term in a geometric series?
A: The first term is the first number in the series.
Q: How do I find the first term in a geometric series?
A: To find the first term, you can look at the series and identify the first number. For example, if the series is 2, 6, 18, 54, ..., the first term is 2.
Q: What is the number of terms in a geometric series?
A: The number of terms is the total number of terms in the series.
Q: How do I find the number of terms in a geometric series?
A: To find the number of terms, you can count the number of terms in the series. For example, if the series is 2, 6, 18, 54, ..., the number of terms is 4.
Q: What is the sum of the geometric series 2, 6, 18, 54, ...?
A: To find the sum, you can use the formula above. The first term is 2, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 2 * (1 - 3^4) / (1 - 3) S = 2 * (1 - 81) / (-2) S = 2 * (-80) / (-2) S = 80
Q: What is the sum of the geometric series 1, 3, 9, 27, ...?
A: To find the sum, you can use the formula above. The first term is 1, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 1 * (1 - 3^4) / (1 - 3) S = 1 * (1 - 81) / (-2) S = 1 * (-80) / (-2) S = 40
Q: What is the sum of the geometric series 4, 12, 36, 108, ...?
A: To find the sum, you can use the formula above. The first term is 4, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 4 * (1 - 3^4) / (1 - 3) S = 4 * (1 - 81) / (-2) S = 4 * (-80) / (-2) S = 160
Q: What is the sum of the geometric series 6, 18, 54, 162, ...?
A: To find the sum, you can use the formula above. The first term is 6, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 6 * (1 - 3^4) / (1 - 3) S = 6 * (1 - 81) / (-2) S = 6 * (-80) / (-2) S = 240
Q: What is the sum of the geometric series 8, 24, 72, 216, ...?
A: To find the sum, you can use the formula above. The first term is 8, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 8 * (1 - 3^4) / (1 - 3) S = 8 * (1 - 81) / (-2) S = 8 * (-80) / (-2) S = 320
Q: What is the sum of the geometric series 10, 30, 90, 270, ...?
A: To find the sum, you can use the formula above. The first term is 10, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 10 * (1 - 3^4) / (1 - 3) S = 10 * (1 - 81) / (-2) S = 10 * (-80) / (-2) S = 400
Q: What is the sum of the geometric series 12, 36, 108, 324, ...?
A: To find the sum, you can use the formula above. The first term is 12, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 12 * (1 - 3^4) / (1 - 3) S = 12 * (1 - 81) / (-2) S = 12 * (-80) / (-2) S = 480
Q: What is the sum of the geometric series 14, 42, 126, 378, ...?
A: To find the sum, you can use the formula above. The first term is 14, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 14 * (1 - 3^4) / (1 - 3) S = 14 * (1 - 81) / (-2) S = 14 * (-80) / (-2) S = 560
Q: What is the sum of the geometric series 16, 48, 144, 432, ...?
A: To find the sum, you can use the formula above. The first term is 16, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 16 * (1 - 3^4) / (1 - 3) S = 16 * (1 - 81) / (-2) S = 16 * (-80) / (-2) S = 640
Q: What is the sum of the geometric series 18, 54, 162, 486, ...?
A: To find the sum, you can use the formula above. The first term is 18, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 18 * (1 - 3^4) / (1 - 3) S = 18 * (1 - 81) / (-2) S = 18 * (-80) / (-2) S = 720
Q: What is the sum of the geometric series 20, 60, 180, 540, ...?
A: To find the sum, you can use the formula above. The first term is 20, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 20 * (1 - 3^4) / (1 - 3) S = 20 * (1 - 81) / (-2) S = 20 * (-80) / (-2) S = 800
Q: What is the sum of the geometric series 22, 66, 198, 594, ...?
A: To find the sum, you can use the formula above. The first term is 22, the common ratio is 3, and the number of terms is 4. Plugging these values into the formula, you get:
S = 22 * (1 - 3^4) / (1 - 3) S = 22 * (1 - 81) / (-2) S = 22 * (-80) / (-2) S = 880
Q: What is the sum of the geometric series 24, 72, 216, 648, ...?
A: To find the sum, you can use the formula above.