Select The Correct Answer.What Is This Series Written In Sigma Notation?${ 2.5 + 2.5(1.2) + 2.5(1.2)^2 + \cdots + 2.5(1.2)^{87} }$A. { \sum_{k=1}^{87} 2.5(1.2)^k$}$B. { \sum_{k=1}^{87} 2.5(1.2)^{k-1}$}$C.

Introduction

Sigma notation is a powerful tool used in mathematics to represent the sum of a series of numbers. It is commonly used in calculus, algebra, and other branches of mathematics to simplify complex expressions and make them easier to work with. In this article, we will explore the concept of sigma notation and how it is used to represent the given series.

What is Sigma Notation?

Sigma notation is a shorthand way of writing the sum of a series of numbers. It is represented by the Greek letter sigma, which is the 18th letter of the Greek alphabet. The sigma notation is used to represent the sum of a series of numbers, where each number is multiplied by a common factor.

The General Form of Sigma Notation

The general form of sigma notation is:

$$\sum_{k=1}^{n} a_k$$

Where:

• $\sum$ is the sigma symbol
• $k$ is the index of the series
• $a_k$ is the general term of the series
• $n$ is the number of terms in the series

Representing the Given Series in Sigma Notation

The given series is:

$$2.5 + 2.5(1.2) + 2.5(1.2)^2 + \cdots + 2.5(1.2)^{87}$$

To represent this series in sigma notation, we need to identify the general term and the number of terms. The general term is $2.5(1.2)^k$, and the number of terms is 87.

Option A: $\sum_{k=1}^{87} 2.5(1.2)^k$

This option represents the series as:

$$\sum_{k=1}^{87} 2.5(1.2)^k$$

This is a correct representation of the series, as it includes the general term $2.5(1.2)^k$ and the number of terms 87.

Option B: $\sum_{k=1}^{87} 2.5(1.2)^{k-1}$

This option represents the series as:

$$\sum_{k=1}^{87} 2.5(1.2)^{k-1}$$

This is not a correct representation of the series, as it includes the general term $2.5(1.2)^{k-1}$, which is not the same as the original series.

Conclusion

In conclusion, the correct representation of the given series in sigma notation is:

$$\sum_{k=1}^{87} 2.5(1.2)^k$$

This representation includes the general term $2.5(1.2)^k$ and the number of terms 87, making it a correct and accurate representation of the series.

Discussion

The concept of sigma notation is a powerful tool used in mathematics to represent the sum of a series of numbers. It is commonly used in calculus, algebra, and other branches of mathematics to simplify complex expressions and make them easier to work with. In this article, we explored the concept of sigma notation and how it is used to represent the given series.

Key Takeaways

• Sigma notation is a shorthand way of writing the sum of a series of numbers.
• The general form of sigma notation is $\sum_{k=1}^{n} a_k$.
• The given series can be represented in sigma notation as $\sum_{k=1}^{87} 2.5(1.2)^k$.
• The correct representation of the series includes the general term $2.5(1.2)^k$ and the number of terms 87.

Further Reading

For further reading on sigma notation and its applications in mathematics, we recommend the following resources:

• "Sigma Notation" by Math Is Fun
• "Sigma Notation" by Khan Academy
• "Sigma Notation" by Wolfram MathWorld

References

• "Calculus" by Michael Spivak
• "Algebra" by Michael Artin
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Sigma Notation Q&A =====================

Frequently Asked Questions About Sigma Notation

Q: What is sigma notation?

A: Sigma notation is a shorthand way of writing the sum of a series of numbers. It is represented by the Greek letter sigma, which is the 18th letter of the Greek alphabet.

Q: What is the general form of sigma notation?

A: The general form of sigma notation is:

$$\sum_{k=1}^{n} a_k$$

Where:

• $\sum$ is the sigma symbol
• $k$ is the index of the series
• $a_k$ is the general term of the series
• $n$ is the number of terms in the series

Q: How do I read sigma notation?

A: To read sigma notation, you need to understand the following components:

• The sigma symbol $\sum$
• The index of the series $k$
• The general term of the series $a_k$
• The number of terms in the series $n$

For example, the sigma notation $\sum_{k=1}^{5} k^2$ can be read as "the sum of the squares of the first 5 positive integers".

Q: What is the difference between sigma notation and other notations?

A: Sigma notation is different from other notations in that it represents the sum of a series of numbers. Other notations, such as product notation, represent the product of a series of numbers.

Q: How do I use sigma notation to represent a series?

A: To use sigma notation to represent a series, you need to follow these steps:

1. Identify the general term of the series.
2. Identify the number of terms in the series.
3. Write the sigma notation using the general term and the number of terms.

For example, the series $2 + 4 + 6 + \cdots + 20$ can be represented in sigma notation as $\sum_{k=1}^{10} 2k$.

Q: Can I use sigma notation to represent a series with a variable index?

A: Yes, you can use sigma notation to represent a series with a variable index. For example, the series $\sum_{k=1}^{n} k^2$ represents the sum of the squares of the first $n$ positive integers.

Q: How do I evaluate a sigma notation expression?

A: To evaluate a sigma notation expression, you need to follow these steps:

1. Evaluate the general term of the series.
2. Evaluate the number of terms in the series.
3. Substitute the values into the sigma notation expression.

For example, the sigma notation expression $\sum_{k=1}^{5} k^2$ can be evaluated as follows:

• Evaluate the general term: $k^2 = 1^2 = 1$
• Evaluate the number of terms: $n = 5$
• Substitute the values into the sigma notation expression: $\sum_{k=1}^{5} 1 = 1 + 1 + 1 + 1 + 1 = 5$

Q: Can I use sigma notation to represent a series with a fractional index?

A: Yes, you can use sigma notation to represent a series with a fractional index. For example, the series $\sum_{k=1/2}^{5/2} k^2$ represents the sum of the squares of the first 5 positive integers with a fractional index.

Q: How do I use sigma notation to represent a series with a negative index?

A: Yes, you can use sigma notation to represent a series with a negative index. For example, the series $\sum_{k=-1}^{0} k^2$ represents the sum of the squares of the first 2 negative integers.

Conclusion

Sigma notation is a powerful tool used in mathematics to represent the sum of a series of numbers. It is commonly used in calculus, algebra, and other branches of mathematics to simplify complex expressions and make them easier to work with. In this article, we explored the concept of sigma notation and answered some frequently asked questions about it.

Key Takeaways

• Sigma notation is a shorthand way of writing the sum of a series of numbers.
• The general form of sigma notation is $\sum_{k=1}^{n} a_k$.
• Sigma notation can be used to represent a series with a variable index, a fractional index, and a negative index.
• Sigma notation can be used to evaluate a series by substituting the values into the sigma notation expression.

Further Reading

For further reading on sigma notation and its applications in mathematics, we recommend the following resources:

• "Sigma Notation" by Math Is Fun
• "Sigma Notation" by Khan Academy
• "Sigma Notation" by Wolfram MathWorld

References

• "Calculus" by Michael Spivak
• "Algebra" by Michael Artin
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton