Identifying the Given Sequence

Oct 13, 2025 by talynt.org 21 views

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 finite geometric series can be calculated using a specific formula, which is essential in various mathematical and real-world applications. In this article, we will explore how to find the sum of the first 7 terms of a given geometric sequence using the formula for the sum of a finite geometric series.

Understanding the Formula

The formula for the sum of a finite geometric series is given by:

${ S_n = \frac{a_1 - a_1 r^n}{1 - r} }$

where:

${ S_n }$ is the sum of the first n terms of the series
${ a_1 }$ is the first term of the series
${ r }$ is the common ratio
${ n }$ is the number of terms

Identifying the Given Sequence

The given sequence is:

${ 18, \quad -6, \quad 2, \ldots }$

We can see that the first term, ${ a_1 }$ , is 18. To find the common ratio, ${ r }$ , we can divide the second term by the first term:

${ r = \frac{-6}{18} = -\frac{1}{3} }$

Applying the Formula

Now that we have identified the first term and the common ratio, we can apply the formula to find the sum of the first 7 terms:

${ S_7 = \frac{18 - 18 \left(-\frac{1}{3}\right)^7}{1 - \left(-\frac{1}{3}\right)} }$

To simplify the expression, we can calculate the value of ${ \left(-\frac{1}{3}\right)^7 }$ :

${ \left(-\frac{1}{3}\right)^7 = -\frac{1}{2187} }$

Now, we can substitute this value back into the formula:

${ S_7 = \frac{18 - 18 \left(-\frac{1}{2187}\right)}{1 - \left(-\frac{1}{3}\right)} }$

${ S_7 = \frac{18 + \frac{18}{2187}}{\frac{4}{3}} }$

${ S_7 = \frac{\frac{18 \cdot 2187 + 18}{2187}}{\frac{4}{3}} }$

${ S_7 = \frac{\frac{39366 + 18}{2187}}{\frac{4}{3}} }$

${ S_7 = \frac{\frac{39384}{2187}}{\frac{4}{3}} }$

${ S_7 = \frac{39384}{2187} \cdot \frac{3}{4} }$

${ S_7 = \frac{39384 \cdot 3}{2187 \cdot 4} }$

${ S_7 = \frac{118152}{8758} }$

${ S_7 = 13.5 }$

Conclusion

In this article, we have used the formula for the sum of a finite geometric series to find the sum of the first 7 terms of a given sequence. We have identified the first term and the common ratio, and then applied the formula to calculate the sum. The result is 13.5, which is the sum of the first 7 terms of the given sequence.

Discussion

The formula for the sum of a finite geometric series is a powerful tool in mathematics, and it has many real-world applications. It can be used to calculate the sum of a sequence of numbers, which is essential in various fields such as finance, economics, and engineering. In this article, we have demonstrated how to use the formula to find the sum of a given sequence, and we have provided a step-by-step guide on how to apply the formula.

References

[1] "Geometric Series" by Math Open Reference. Retrieved from https://www.mathopenref.com/geomseries.html
[2] "Sum of a Finite Geometric Series" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/FiniteGeometricSeries.html

Additional Resources

[1] "Geometric Series Calculator" by Calculator Soup. Retrieved from https://www.calculatorsoup.com/calculators/series-geometric.php
[2] "Sum of a Finite Geometric Series Formula" by Mathway. Retrieved from https://www.mathway.com/answers/series/finite_geometric_series/sum_formula.html
Q&A: Finding the Sum of a Finite Geometric Series =====================================================

Introduction

In our previous article, we explored how to find the sum of a finite geometric series using the formula:

${ S_n = \frac{a_1 - a_1 r^n}{1 - r} }$

We applied this formula to a given sequence and calculated the sum of the first 7 terms. In this article, we will answer some frequently asked questions related to finding the sum of a finite 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 one by a fixed, non-zero number called the common ratio.

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:

${ S_n = \frac{a_1 - a_1 r^n}{1 - r} }$

Q: What is the first term (a1) in the formula?

A: The first term (a1) is the first number in the sequence.

Q: What is the common ratio (r) in the formula?

A: The common ratio (r) is the number by which each term is multiplied to get the next term.

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

A: To calculate the sum of a finite geometric series, you need to:

Identify the first term (a1) and the common ratio (r)
Plug these values into the formula
Simplify the expression to get the sum

Q: What if the common ratio is negative?

A: If the common ratio is negative, the formula will still work. However, you need to be careful when simplifying the expression, as the negative sign may affect the result.

Q: Can I use the formula for the sum of a finite geometric series for any sequence?

A: No, the formula is only applicable to geometric sequences, where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Q: What if I want to find the sum of a sequence with a large number of terms?

A: In that case, you can use a calculator or a computer program to calculate the sum. Alternatively, you can use the formula for the sum of an infinite geometric series, which is:

${ S = \frac{a_1}{1 - r} }$

However, this formula is only applicable if the absolute value of the common ratio (|r|) is less than 1.

Q: Can I use the formula for the sum of a finite geometric series for real-world applications?

A: Yes, the formula has many real-world applications, such as:

Calculating the future value of an investment
Determining the total cost of a project
Finding the sum of a sequence of numbers in finance, economics, and engineering

Conclusion

In this article, we have answered some frequently asked questions related to finding the sum of a finite geometric series. We have provided explanations and examples to help you understand the formula and its applications. Whether you are a student or a professional, the formula for the sum of a finite geometric series is a powerful tool that can help you solve problems and make informed decisions.

Discussion

The formula for the sum of a finite geometric series is a fundamental concept in mathematics, and it has many real-world applications. In this article, we have demonstrated how to use the formula to find the sum of a given sequence, and we have provided a step-by-step guide on how to apply the formula.

References

[1] "Geometric Series" by Math Open Reference. Retrieved from https://www.mathopenref.com/geomseries.html
[2] "Sum of a Finite Geometric Series" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/FiniteGeometricSeries.html

Additional Resources

[1] "Geometric Series Calculator" by Calculator Soup. Retrieved from https://www.calculatorsoup.com/calculators/series-geometric.php
[2] "Sum of a Finite Geometric Series Formula" by Mathway. Retrieved from https://www.mathway.com/answers/series/finite_geometric_series/sum_formula.html