QUESTION 11.1 Given: $ S_n = \frac{a(r^n - 1)}{r - 1} $1.1.1 Write $ S_n $ In Expanded Form.1.1.2 Which Term In 1.1.1 Is Affected By The Number Of Terms In The Series?Consider $ R^n $ (write The Answer In Scientific Notation

Introduction

In mathematics, a geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant. The sum of a geometric series can be calculated using a formula, which is given by $ S_n = \frac{a(r^n - 1)}{r - 1} $. In this article, we will explore the expanded form of this formula and identify which term is affected by the number of terms in the series.

Expanded Form of the Formula

To write the formula in expanded form, we need to expand the expression $ r^n - 1 $. This can be done using the formula for the difference of squares, which is $ a^2 - b^2 = (a - b)(a + b) $. In this case, we have $ r^n - 1 = (r - 1)(r^{n-1} + r^{n-2} + \ldots + r + 1) $. Substituting this expression into the original formula, we get:

$$S_n = \frac{a(r - 1)(r^{n-1} + r^{n-2} + \ldots + r + 1)}{r - 1}$$

Simplifying this expression, we get:

$$S_n = a(r^{n-1} + r^{n-2} + \ldots + r + 1)$$

This is the expanded form of the formula for the sum of a geometric series.

Term Affected by the Number of Terms

The term in the expanded form that is affected by the number of terms in the series is $ r^{n-1} $. This term is raised to the power of $ n-1 $, which means that it is affected by the number of terms in the series. As the number of terms increases, the value of $ r^{n-1} $ also increases.

Scientific Notation

To write $ r^n $ in scientific notation, we need to express it in the form $ a \times 10^b $, where $ a $ is a number between 1 and 10, and $ b $ is an integer. In this case, we have:

$$r^n = r \times r^{n-1}$$

Using the fact that $ r^{n-1} = (r^{n-1})1 $, we can write:

$$r^n = r \times (r^{n-1})^1$$

Using the property of exponents that $ (a^m)n = a^{mn} $, we can rewrite this as:

$$r^n = r \times r^{(n-1) \times 1}$$

Simplifying this expression, we get:

$$r^n = r^{n}$$

This is the scientific notation for $ r^n $.

Conclusion

In this article, we have explored the expanded form of the formula for the sum of a geometric series and identified which term is affected by the number of terms in the series. We have also written $ r^n $ in scientific notation. The formula for the sum of a geometric series is a powerful tool for calculating the sum of a series, and understanding its expanded form and the term affected by the number of terms is essential for applying it in various mathematical and real-world problems.

References

• [1] "Geometric Series" by Math Open Reference
• [2] "Sum of a Geometric Series" by Wolfram MathWorld

Further Reading

• [1] "Geometric Series and Sequences" by Khan Academy
• [2] "Sum of a Geometric Series" by MIT OpenCourseWare
  Frequently Asked Questions (FAQs) about Geometric Series ===========================================================

Q: What is a geometric series?

A: A geometric series is a type of series where each term is obtained by multiplying the previous term by a fixed constant. The series has the form $ a, ar, ar^2, ar^3, \ldots, ar^{n-1} $, where $ a $ is the first term and $ r $ is 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 given by $ S_n = \frac{a(r^n - 1)}{r - 1} $, where $ S_n $ is the sum of the first $ n $ terms, $ a $ is the first term, $ r $ is the common ratio, and $ n $ is the number of terms.

Q: What is the expanded form of the formula for the sum of a geometric series?

A: The expanded form of the formula for the sum of a geometric series is given by $ S_n = a(r^{n-1} + r^{n-2} + \ldots + r + 1) $.

Q: Which term in the expanded form is affected by the number of terms in the series?

A: The term $ r^{n-1} $ in the expanded form is affected by the number of terms in the series.

Q: How do I write $ r^n $ in scientific notation?

A: To write $ r^n $ in scientific notation, you can express it in the form $ a \times 10^b $, where $ a $ is a number between 1 and 10, and $ b $ is an integer.

Q: What is the significance of the common ratio $ r $ in a geometric series?

A: The common ratio $ r $ determines the rate at which the terms of the series increase or decrease. If $ r > 1 $, the terms increase; if $ r < 1 $, the terms decrease; and if $ r = 1 $, the series is constant.

Q: Can a geometric series have a negative common ratio?

A: Yes, a geometric series can have a negative common ratio. In this case, the terms of the series will alternate in sign.

Q: How do I calculate the sum of a geometric series with a negative common ratio?

A: To calculate the sum of a geometric series with a negative common ratio, you can use the formula $ S_n = \frac{a(r^n - 1)}{r - 1} $, where $ r $ is the negative common ratio.

Q: What is the relationship between the sum of a geometric series and the number of terms?

A: The sum of a geometric series is directly proportional to the number of terms. As the number of terms increases, the sum of the series also increases.

Q: Can a geometric series have a zero common ratio?

A: No, a geometric series cannot have a zero common ratio. If $ r = 0 $, the series is not geometric.

Q: How do I determine if a series is geometric?

A: To determine if a series is geometric, you can check if each term is obtained by multiplying the previous term by a fixed constant. If this is the case, the series is geometric.

Q: What are some real-world applications of geometric series?

A: Geometric series have many real-world applications, including finance, economics, and engineering. They are used to model population growth, compound interest, and electrical circuits.

Conclusion

In this article, we have answered some frequently asked questions about geometric series. We have discussed the formula for the sum of a geometric series, the expanded form of the formula, and the significance of the common ratio. We have also explored some real-world applications of geometric series.