The Sum Of A Geometric Series With 15 Terms, Where The First Term Is 120 And The Common Ratio Is 0.85, Is Closest To Which Of The Following?1. 730 2. 745 3. 820 4. 910
Introduction
In mathematics, a geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The sum of a geometric series is a fundamental concept in mathematics, with numerous applications in finance, engineering, and other fields. In this article, we will explore the sum of a geometric series with 15 terms, where the first term is 120 and the common ratio is 0.85.
What is a Geometric Series?
A geometric series is a sequence of numbers that can be represented as:
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 * (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.
Calculating the Sum of a Geometric Series
To calculate the sum of a geometric series, we need to plug in the values of 'a', 'r', and 'n' into the formula. In this case, the first term 'a' is 120, the common ratio 'r' is 0.85, and the number of terms 'n' is 15.
S = 120 * (1 - 0.85^15) / (1 - 0.85)
Using a Calculator to Find the Sum
To find the sum of the series, we can use a calculator to evaluate the expression:
S = 120 * (1 - 0.85^15) / (1 - 0.85)
Using a calculator, we get:
S ≈ 745.31
Rounding the Sum
Since the options are given as integers, we need to round the sum to the nearest integer. Rounding 745.31 to the nearest integer gives us 745.
Conclusion
In conclusion, the sum of a geometric series with 15 terms, where the first term is 120 and the common ratio is 0.85, is closest to 745.
Comparison with Other Options
Let's compare our result with the other options:
- Option 1: 730
- Option 2: 745
- Option 3: 820
- Option 4: 910
Our result, 745, is closest to option 2, 745.
Real-World Applications
The sum of a geometric series has numerous real-world applications, including:
- Finance: The sum of a geometric series can be used to calculate the future value of an investment or a loan.
- Engineering: The sum of a geometric series can be used to calculate the total amount of energy transferred in a system.
- Computer Science: The sum of a geometric series can be used to calculate the total amount of memory required to store a large dataset.
Conclusion
In conclusion, the sum of a geometric series with 15 terms, where the first term is 120 and the common ratio is 0.85, is closest to 745. The sum of a geometric series has numerous real-world applications, including finance, engineering, and computer science.
References
- Mathematics Handbook: A comprehensive guide to mathematical formulas and concepts.
- Geometric Series: A detailed explanation of geometric series and their applications.
- Calculator: A tool used to evaluate mathematical expressions.
Further Reading
- Geometric Series Formula: A detailed explanation of the formula for the sum of a geometric series.
- Geometric Series Examples: A collection of examples illustrating the use of geometric series in real-world applications.
- Geometric Series Calculator: A tool used to calculate the sum of a geometric series.
The Sum of a Geometric Series: A Q&A Guide =====================================================
Introduction
In our previous article, we explored the sum of a geometric series with 15 terms, where the first term is 120 and the common ratio is 0.85. In this article, we will answer some frequently asked questions about geometric series and their sums.
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 one 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 = 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: What is the common ratio?
A: The common ratio is a fixed, non-zero number that is used to multiply each term in a geometric series to get the next term.
Q: How do I find the common ratio?
A: The common ratio can be found by dividing any term in the series by the previous term.
Q: What is the first term?
A: The first term is the first number in the geometric series.
Q: How do I find the first term?
A: The first term is given in the problem statement.
Q: What is the number of terms?
A: The number of terms is the total number of terms in the geometric series.
Q: How do I find the number of terms?
A: The number of terms is given in the problem statement.
Q: Can I use a calculator to find the sum of a geometric series?
A: Yes, you can use a calculator to find the sum of a geometric series.
Q: What if the common ratio is greater than 1?
A: If the common ratio is greater than 1, the sum of the geometric series will be infinite.
Q: What if the common ratio is less than 1?
A: If the common ratio is less than 1, the sum of the geometric series will be finite.
Q: Can I use the sum of a geometric series to calculate the future value of an investment?
A: Yes, you can use the sum of a geometric series to calculate the future value of an investment.
Q: Can I use the sum of a geometric series to calculate the total amount of energy transferred in a system?
A: Yes, you can use the sum of a geometric series to calculate the total amount of energy transferred in a system.
Conclusion
In conclusion, the sum of a geometric series is a fundamental concept in mathematics with numerous real-world applications. By understanding how to calculate the sum of a geometric series, you can apply it to a wide range of problems in finance, engineering, and other fields.
References
- Mathematics Handbook: A comprehensive guide to mathematical formulas and concepts.
- Geometric Series: A detailed explanation of geometric series and their applications.
- Calculator: A tool used to evaluate mathematical expressions.
Further Reading
- Geometric Series Formula: A detailed explanation of the formula for the sum of a geometric series.
- Geometric Series Examples: A collection of examples illustrating the use of geometric series in real-world applications.
- Geometric Series Calculator: A tool used to calculate the sum of a geometric series.