Writing the Series using Sigma Notation

Oct 13, 2025 by windailys.com 100 views

Introduction

Sigma notation is a powerful tool used in mathematics to represent the sum of a series. It is a compact and concise way to express the sum of a sequence of numbers. In this article, we will explore how to write the series $3 + 12 + 48 + 192$ using sigma notation.

What is Sigma Notation?

Sigma notation is a mathematical notation that represents the sum of a series. It is denoted by the Greek letter sigma, $\sum$ . The general form of sigma notation is:

\sum_{i=1}^{n} a_i

where $a_i$ is the $i^{th}$ term of the series, and $n$ is the number of terms.

Writing the Series using Sigma Notation

To write the series $3 + 12 + 48 + 192$ using sigma notation, we need to identify the pattern of the series. The series can be written as:

3 + 12 + 48 + 192

We can see that each term is obtained by multiplying 3 by a power of 4. Specifically, the first term is $3 \cdot 4^0$ , the second term is $3 \cdot 4^1$ , the third term is $3 \cdot 4^2$ , and the fourth term is $3 \cdot 4^3$ .

Using the Formula for Sigma Notation

Using the formula for sigma notation, we can write the series as:

\sum_{n=1}^{4} (3)(4)^{n-1}

This is because the first term is $3 \cdot 4^0$ , the second term is $3 \cdot 4^1$ , the third term is $3 \cdot 4^2$ , and the fourth term is $3 \cdot 4^3$ .

Evaluating the Series

To evaluate the series, we can use the formula for the sum of a geometric series:

\sum_{n=1}^{n} ar^{n-1} = \frac{a(1-r^n)}{1-r}

In this case, $a = 3$ , $r = 4$ , and $n = 4$ . Plugging these values into the formula, we get:

\sum_{n=1}^{4} (3)(4)^{n-1} = \frac{3(1-4^4)}{1-4}

Simplifying the expression, we get:

\sum_{n=1}^{4} (3)(4)^{n-1} = \frac{3(1-256)}{-3}

\sum_{n=1}^{4} (3)(4)^{n-1} = \frac{3(-255)}{-3}

\sum_{n=1}^{4} (3)(4)^{n-1} = 255

Conclusion

In this article, we have seen how to write the series $3 + 12 + 48 + 192$ using sigma notation. We have also evaluated the series using the formula for the sum of a geometric series. The correct answer is:

\sum_{n=1}^{4} (3)(4)^{n-1}

This is the only option that correctly represents the series using sigma notation.

Answer Key

A. $\sum_{n=1}^4 (3)(-4)^{n-1}$

B. $\sum_{n=1}^3 (3)(4)^{n-1}$

C. $\sum_{n=1}^4 (4)(3)^{n-1}$

D. $\sum_{n=1}^4 (3)(4)^{n-1}$

Q&A: Sigma Notation and Series

Q: What is sigma notation?

A: Sigma notation is a mathematical notation that represents the sum of a series. It is denoted by the Greek letter sigma, $\sum$ . The general form of sigma notation is:

\sum_{i=1}^{n} a_i

where $a_i$ is the $i^{th}$ term of the series, and $n$ is the number of terms.

Q: How do I write a series using sigma notation?

A: To write a series using sigma notation, you need to identify the pattern of the series. You can then use the formula for sigma notation to write the series. For example, if the series is $3 + 12 + 48 + 192$ , you can write it as:

\sum_{n=1}^{4} (3)(4)^{n-1}

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

A: The formula for the sum of a geometric series is:

\sum_{n=1}^{n} ar^{n-1} = \frac{a(1-r^n)}{1-r}

where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I evaluate a series using sigma notation?

A: To evaluate a series using sigma notation, you can use the formula for the sum of a geometric series. For example, if the series is $\sum_{n=1}^{4} (3)(4)^{n-1}$ , you can evaluate it as:

\sum_{n=1}^{4} (3)(4)^{n-1} = \frac{3(1-4^4)}{1-4}

Simplifying the expression, you get:

\sum_{n=1}^{4} (3)(4)^{n-1} = 255

Q: What are some common mistakes to avoid when using sigma notation?

A: Some common mistakes to avoid when using sigma notation include:

Not identifying the pattern of the series
Not using the correct formula for the sum of a geometric series
Not simplifying the expression correctly
Not checking the validity of the series

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

A: To determine if a series is valid, you need to check if the series converges. A series converges if the limit of the sum of the terms as the number of terms approaches infinity is finite. You can use the formula for the sum of a geometric series to check if a series converges.

Q: What are some real-world applications of sigma notation?

A: Sigma notation has many real-world applications, including:

Calculating the sum of a series of numbers
Modeling population growth
Calculating the sum of a series of financial transactions
Modeling the spread of a disease

Conclusion

In this article, we have seen how to write a series using sigma notation, how to evaluate a series using sigma notation, and how to determine if a series is valid. We have also seen some common mistakes to avoid when using sigma notation and some real-world applications of sigma notation.

Answer Key

Q: What is sigma notation? A: Sigma notation is a mathematical notation that represents the sum of a series.

Q: How do I write a series using sigma notation? A: To write a series using sigma notation, you need to identify the pattern of the series and use the formula for sigma notation.

Q: What is the formula for the sum of a geometric series? A: The formula for the sum of a geometric series is $\sum_{n=1}^{n} ar^{n-1} = \frac{a(1-r^n)}{1-r}$ .

Q: How do I evaluate a series using sigma notation? A: To evaluate a series using sigma notation, you can use the formula for the sum of a geometric series.

Q: What are some common mistakes to avoid when using sigma notation? A: Some common mistakes to avoid when using sigma notation include not identifying the pattern of the series, not using the correct formula for the sum of a geometric series, not simplifying the expression correctly, and not checking the validity of the series.

Q: How do I determine if a series is valid? A: To determine if a series is valid, you need to check if the series converges.

Q: What are some real-world applications of sigma notation? A: Sigma notation has many real-world applications, including calculating the sum of a series of numbers, modeling population growth, calculating the sum of a series of financial transactions, and modeling the spread of a disease.