Calculating the General Term of the Series

Oct 13, 2025 by recoveryshake.com 42 views

Introduction

In this article, we will delve into the world of mathematics and explore a series of numbers that follow a specific pattern. The series in question is: $$1+5+9+\ldots+97$$ We will break down the series into smaller, manageable parts and calculate the general term, determine the number of terms, and express the series in sigma notation.

Calculating the General Term of the Series

To find the general term of the series, we need to identify the pattern and express it mathematically. The series starts with 1 and increases by 4 each time. This means that the second term is 1 + 4 = 5, the third term is 5 + 4 = 9, and so on.

Let's denote the general term as $a_n$ , where $n$ is the term number. We can express the general term as:

a_n = 1 + 4(n-1)

This equation represents the $n$ th term of the series. We can simplify it by expanding the equation:

a_n = 1 + 4n - 4

a_n = 4n - 3

This is the general term of the series.

Calculating the Number of Terms in the Series

To determine the number of terms in the series, we need to find the value of $n$ when the last term is 97. We can set up an equation using the general term:

a_n = 97

4n - 3 = 97

4n = 100

n = 25

This means that there are 25 terms in the series.

Expressing the Series in Sigma Notation

Sigma notation is a compact way to express a series. It consists of three parts: the index of summation, the lower limit, and the upper limit. The general term of the series is $a_n = 4n - 3$ . We can express the series in sigma notation as:

\sum_{n=1}^{25} (4n - 3)

This equation represents the series in sigma notation.

Simplifying the Series

We can simplify the series by expanding the sigma notation:

\sum_{n=1}^{25} (4n - 3) = 4\sum_{n=1}^{25} n - 3\sum_{n=1}^{25} 1

We can evaluate the two summations separately:

\sum_{n=1}^{25} n = \frac{25(25+1)}{2} = 325

\sum_{n=1}^{25} 1 = 25

Substituting these values back into the equation, we get:

4(325) - 3(25) = 1300 - 75 = 1225

This is the sum of the series.

Conclusion

In this article, we explored a series of numbers that follow a specific pattern. We calculated the general term of the series, determined the number of terms, and expressed the series in sigma notation. We also simplified the series by expanding the sigma notation and evaluating the two summations. The sum of the series is 1225.

Further Exploration

This series can be extended to include more terms. We can calculate the general term for a larger number of terms and express the series in sigma notation. We can also explore other mathematical concepts, such as the arithmetic series and the geometric series.

References

[1] "Arithmetic Series" by Math Open Reference
[2] "Geometric Series" by Math Open Reference

Glossary

Arithmetic Series: A series of numbers in which each term is obtained by adding a fixed constant to the previous term.
Geometric Series: A series of numbers in which each term is obtained by multiplying the previous term by a fixed constant.
Sigma Notation: A compact way to express a series, consisting of three parts: the index of summation, the lower limit, and the upper limit.
Frequently Asked Questions: Understanding the Series =====================================================

Q: What is the pattern of the series?

A: The series starts with 1 and increases by 4 each time. This means that the second term is 1 + 4 = 5, the third term is 5 + 4 = 9, and so on.

Q: How do I calculate the general term of the series?

A: To find the general term of the series, you can use the equation: $$a_n = 1 + 4(n-1)$$ This equation represents the $n$ th term of the series.

Q: How many terms are in the series?

A: To determine the number of terms in the series, you can set up an equation using the general term: $$a_n = 97$$ $$4n - 3 = 97$$ $$4n = 100$$ $$n = 25$$ This means that there are 25 terms in the series.

Q: How do I express the series in sigma notation?

A: You can express the series in sigma notation as: $$\sum_{n=1}^{25} (4n - 3)$$ This equation represents the series in sigma notation.

Q: What is the sum of the series?

A: To find the sum of the series, you can simplify the sigma notation by expanding it: $$\sum_n=1}^{25} (4n - 3) = 4\sum_{n=1}^{25} n - 3\sum_{n=1}^{25} 1$$ Evaluating the two summations separately, you get $$\sum_{n=1^{25} n = \frac{25(25+1)}{2} = 325$$ $$\sum_{n=1}^{25} 1 = 25$$ Substituting these values back into the equation, you get: $$4(325) - 3(25) = 1300 - 75 = 1225$$ This is the sum of the series.

Q: Can I extend the series to include more terms?

A: Yes, you can extend the series to include more terms. You can calculate the general term for a larger number of terms and express the series in sigma notation.

Q: What are some other mathematical concepts related to this series?

A: Some other mathematical concepts related to this series include the arithmetic series and the geometric series.

Q: Where can I learn more about the arithmetic series and the geometric series?

A: You can learn more about the arithmetic series and the geometric series by visiting the following resources:

[1] "Arithmetic Series" by Math Open Reference
[2] "Geometric Series" by Math Open Reference

Q: What is the difference between the arithmetic series and the geometric series?

A: The arithmetic series is a series of numbers in which each term is obtained by adding a fixed constant to the previous term. The geometric series is a series of numbers in which each term is obtained by multiplying the previous term by a fixed constant.

Q: Can I use the series in real-world applications?

A: Yes, you can use the series in real-world applications. For example, you can use the series to model population growth or to calculate the total cost of a project.

Q: Where can I find more resources on the series?

A: You can find more resources on the series by visiting the following websites:

[1] Math Open Reference
[2] Khan Academy
[3] Wolfram Alpha

Conclusion

In this article, we have answered some frequently asked questions about the series. We have covered topics such as the pattern of the series, calculating the general term, expressing the series in sigma notation, and more. We hope that this article has been helpful in understanding the series and its applications.