Express The Sum Of The First 50 Terms Of The Sequence In Sigma Notation: { -1, -\frac{1}{6}, -\frac{1}{36}, \ldots$}$.A. { \sum_{n=1} {50}(-1)(50) {n-1}$}$B. { \sum_{n=1} {50}\left(-\frac{1}{6}\right)(-1) {n-1}$}$C.

Introduction

Sigma notation is a powerful tool used to express the sum of a series in a concise and elegant way. It is commonly used in mathematics, particularly in calculus and number theory, to represent the sum of a sequence. In this article, we will explore how to express the sum of the first 50 terms of a given geometric sequence in sigma notation.

Understanding the Sequence

The given sequence is ${-1, -\frac{1}{6}, -\frac{1}{36}, \ldots}$. This is a geometric sequence, where each term is obtained by multiplying the previous term by a fixed constant. In this case, the common ratio is ${-\frac{1}{6}}$.

Expressing the Sum in Sigma Notation

To express the sum of the first 50 terms of this sequence in sigma notation, we need to identify the first term, the common ratio, and the number of terms. The first term is ${-1}$, the common ratio is ${-\frac{1}{6}}$, and the number of terms is ${50}$.

Option A: Incorrect Expression

Option A is ${\sum_{n=1}^{50}(-1)(50)^{n-1}}$. This expression is incorrect because it does not accurately represent the given sequence. The term ${(-1)(50)^{n-1}}$ does not match the terms in the given sequence.

Option B: Correct Expression

Option B is ${\sum_{n=1}^{50}\left(-\frac{1}{6}\right)(-1)^{n-1}}$. This expression accurately represents the given sequence. The term ${\left(-\frac{1}{6}\right)(-1)^{n-1}}$ matches the terms in the given sequence, and the sum is taken over the first 50 terms.

Derivation of the Correct Expression

To derive the correct expression, we can start by writing out the first few terms of the sequence:

${-1, -\frac{1}{6}, -\frac{1}{36}, \ldots}$

We can see that each term is obtained by multiplying the previous term by ${-\frac{1}{6}}$. Therefore, the nth term of the sequence can be written as:

${\left(-\frac{1}{6}\right)^{n-1}(-1)}$

This is because the first term is ${-1}$, and each subsequent term is obtained by multiplying the previous term by ${-\frac{1}{6}}$.

Conclusion

In conclusion, the correct expression for the sum of the first 50 terms of the given geometric sequence in sigma notation is:

${\sum_{n=1}^{50}\left(-\frac{1}{6}\right)(-1)^{n-1}}$

This expression accurately represents the given sequence and can be used to calculate the sum of the first 50 terms.

References

• [1] "Sigma Notation" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/sigma-notation.html
• [2] "Geometric Sequences" by Purplemath. Retrieved from https://www.purplemath.com/modules/seqgeom.htm

Discussion

Introduction

In our previous article, we explored how to express the sum of the first 50 terms of a given geometric sequence in sigma notation. We derived the correct expression and discussed the importance of accurately representing the sequence. In this article, we will answer some frequently asked questions about expressing the sum of a geometric sequence in sigma notation.

Q: What is a geometric sequence?

A geometric sequence 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 identify the first term, common ratio, and number of terms in a geometric sequence?

To identify the first term, common ratio, and number of terms in a geometric sequence, you can follow these steps:

1. Identify the first term of the sequence.
2. Determine the common ratio by dividing any term by its previous term.
3. Count the number of terms in the sequence.

Q: What is sigma notation?

Sigma notation is a way of expressing the sum of a series in a concise and elegant way. It is commonly used in mathematics, particularly in calculus and number theory, to represent the sum of a sequence.

Q: How do I express the sum of a geometric sequence in sigma notation?

To express the sum of a geometric sequence in sigma notation, you can use the following formula:

${\sum_{n=1}^{N} ar^{n-1}}$

where:

• ${a}$ is the first term of the sequence
• ${r}$ is the common ratio
• ${N}$ is the number of terms in the sequence

Q: What is the correct expression for the sum of the first 50 terms of the given geometric sequence in sigma notation?

The correct expression for the sum of the first 50 terms of the given geometric sequence in sigma notation is:

${\sum_{n=1}^{50}\left(-\frac{1}{6}\right)(-1)^{n-1}}$

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

To calculate the sum of a geometric sequence, you can use the formula:

${S_N = \frac{a(1-r^N)}{1-r}}$

where:

• ${S_N}$ is the sum of the first ${N}$ terms of the sequence
• ${a}$ is the first term of the sequence
• ${r}$ is the common ratio
• ${N}$ is the number of terms in the sequence

Q: What are some common mistakes to avoid when expressing the sum of a geometric sequence in sigma notation?

Some common mistakes to avoid when expressing the sum of a geometric sequence in sigma notation include:

• Failing to identify the first term, common ratio, and number of terms in the sequence
• Using the wrong formula to express the sum of the sequence
• Making errors in the calculation of the sum

Conclusion

In conclusion, expressing the sum of a geometric sequence in sigma notation is an important concept in mathematics. By understanding the formula and common mistakes to avoid, you can accurately represent the sum of a geometric sequence and calculate its value.