Select The Correct Answer.The First Term Of A Geometric Series Is -3, The Common Ratio Is 6, And The Sum Of The Series Is $-4,665$. Using A Table Of Values, How Many Terms Are In This Geometric Series?A. 6 B. 4 C. 10 D. 5
Introduction
Geometric series are a fundamental concept in mathematics, and understanding how to solve problems involving these series is crucial for success in various fields, including finance, engineering, and data analysis. In this article, we will explore how to solve geometric series problems using a table of values, with a focus on finding the number of terms in a given series.
What is a 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 general formula for a geometric series is:
a, ar, ar^2, ar^3, ...
where a is the first term and r is the common ratio.
The Problem
In this problem, we are given the first term (a = -3), the common ratio (r = 6), and the sum of the series (S = -4,665). We need to find the number of terms in this geometric series.
Using a Table of Values
To solve this problem, we can use a table of values to find the number of terms. We will start by listing the first few terms of the series and then use the formula for the sum of a geometric series to find the number of terms.
| Term | Value |
|---|---|
| 1 | -3 |
| 2 | -18 |
| 3 | -108 |
| 4 | -648 |
| 5 | -3888 |
| 6 | -23328 |
Finding the Sum of the Series
The sum of a geometric series can be found using the formula:
S = a * (1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
We can plug in the values we know into this formula to get:
-4,665 = -3 * (1 - 6^n) / (1 - 6)
Simplifying the Equation
To simplify the equation, we can multiply both sides by (1 - 6) to get:
-4,665 * (1 - 6) = -3 * (1 - 6^n)
This simplifies to:
-4,665 * (-5) = -3 * (1 - 6^n)
Solving for n
Now we can solve for n by isolating the term with the exponent:
-23,325 = -3 * (1 - 6^n)
Dividing both sides by -3 gives:
7,775 = 1 - 6^n
Subtracting 1 from both sides gives:
6,775 = -6^n
Dividing both sides by -6 gives:
-1,125.83 = 6^n
Finding the Number of Terms
To find the number of terms, we can use the fact that 6^10 = 60,466,176, which is greater than -1,125.83, and 6^9 = 20,358,521, which is less than -1,125.83. This means that n must be between 9 and 10.
However, we can also use the fact that 6^10 is greater than -1,125.83 to conclude that n must be 10, since 6^9 is less than -1,125.83.
Conclusion
In this article, we have shown how to solve geometric series problems using a table of values. We have also demonstrated how to find the number of terms in a given series by using the formula for the sum of a geometric series. By following these steps, you can solve geometric series problems with ease.
Answer
The correct answer is C. 10.
Discussion
This problem is a great example of how to use a table of values to solve geometric series problems. By listing the first few terms of the series and using the formula for the sum of a geometric series, we can find the number of terms in the series.
Additional Resources
For more information on geometric series, including formulas and examples, see the following resources:
Final Thoughts
Introduction
Geometric series are a fundamental concept in mathematics, and understanding how to solve problems involving these series is crucial for success in various fields. In this article, we will answer some of the most frequently asked questions about geometric series, including how to find the number of terms in a given series, how to use a table of values, and more.
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 find the number of terms in a geometric series?
A: To find the number of terms in a geometric series, you can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
where S is the sum, 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 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, a is the first term, r is the common ratio, and n is the number of terms.
Q: How do I use a table of values to find the number of terms in a geometric series?
A: To use a table of values to find the number of terms in a geometric series, you can list the first few terms of the series and then use the formula for the sum of a geometric series to find the number of terms.
Q: What is the common ratio in a geometric series?
A: The common ratio in a geometric series is the fixed, non-zero number that is multiplied by each term to get the next term.
Q: How do I find the common ratio in a geometric series?
A: To find the common ratio in a geometric series, you can look at the ratio of any two consecutive terms. For example, if the first term is 2 and the second term is 6, the common ratio is 6/2 = 3.
Q: What is the first term in a geometric series?
A: The first term in a geometric series is the first number in the sequence.
Q: How do I find the first term in a geometric series?
A: To find the first term in a geometric series, you can look at the first number in the sequence.
Q: What is the sum of a geometric series?
A: The sum of a geometric series is the total of all the terms in the series.
Q: How do I find the sum of a geometric series?
A: To find the sum of a geometric series, you can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Q: What is the formula for the nth term of a geometric series?
A: The formula for the nth term of a geometric series is:
a_n = a * r^(n-1)
where a_n is the nth term, a is the first term, r is the common ratio, and n is the term number.
Q: How do I find the nth term of a geometric series?
A: To find the nth term of a geometric series, you can use the formula:
a_n = a * r^(n-1)
where a_n is the nth term, a is the first term, r is the common ratio, and n is the term number.
Conclusion
In this article, we have answered some of the most frequently asked questions about geometric series, including how to find the number of terms in a given series, how to use a table of values, and more. By understanding these concepts, you can solve geometric series problems with ease.
Additional Resources
For more information on geometric series, including formulas and examples, see the following resources:
Final Thoughts
Geometric series are a fundamental concept in mathematics, and understanding how to solve problems involving these series is crucial for success in various fields. By following the steps outlined in this article, you can solve geometric series problems with ease. Remember to use a table of values to find the number of terms in a given series, and don't be afraid to use the formula for the sum of a geometric series to help you solve the problem.