Finding the Sum of the First Series

Oct 13, 2025 by miodeya.com 124 views

Introduction

In mathematics, summation is a fundamental concept used to represent the sum of a series of numbers. It is denoted by the Greek letter sigma, and is used to express the sum of a sequence of numbers. In this article, we will explore the concept of summation and how to find the sum of a given series. We will use the formula for the sum of an arithmetic series to find the sum of the given series.

Understanding the Series

The given series is $\sum_{n=1}^{30}(-3n+7)$ . This series represents the sum of the terms of the form $-3n+7$ for $n=1,2,3,...,30$ . To find the sum of this series, we need to first understand the pattern of the terms.

Breaking Down the Series

We can break down the series into two separate series: $\sum_{n=1}^{30}(-3n)$ and $\sum_{n=1}^{30}(7)$ . The first series represents the sum of the terms of the form $-3n$ for $n=1,2,3,...,30$ , and the second series represents the sum of the constant term $7$ for $n=1,2,3,...,30$ .

Finding the Sum of the First Series

The first series is an arithmetic series with a common difference of $-3$ . The formula for the sum of an arithmetic series is given by:

S_n = \frac{n}{2}(a_1 + a_n)

where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the last term.

In this case, the first term is $-3(1)=-3$ and the last term is $-3(30)=-90$ . Plugging these values into the formula, we get:

S_{30} = \frac{30}{2}(-3 + (-90))

S_{30} = 15(-93)

S_{30} = -1395

Finding the Sum of the Second Series

The second series is a constant series with a constant term of $7$ . The formula for the sum of a constant series is given by:

S_n = na

where $S_n$ is the sum of the first $n$ terms, and $a$ is the constant term.

In this case, the constant term is $7$ and the number of terms is $30$ . Plugging these values into the formula, we get:

S_{30} = 30(7)

S_{30} = 210

Finding the Sum of the Original Series

Now that we have found the sum of the two separate series, we can find the sum of the original series by adding the two sums together:

\sum_{n=1}^{30}(-3n+7) = S_{30} + S_{30}

\sum_{n=1}^{30}(-3n+7) = -1395 + 210

\sum_{n=1}^{30}(-3n+7) = -1185

Conclusion

In this article, we explored the concept of summation and how to find the sum of a given series. We used the formula for the sum of an arithmetic series to find the sum of the first series, and the formula for the sum of a constant series to find the sum of the second series. We then added the two sums together to find the sum of the original series. The final answer is $\boxed{-1185}$ .

Frequently Asked Questions

What is the formula for the sum of an arithmetic series? The formula for the sum of an arithmetic series is given by: $S_n = \frac{n}{2}(a_1 + a_n)$
What is the formula for the sum of a constant series? The formula for the sum of a constant series is given by: $S_n = na$
How do I find the sum of a given series? To find the sum of a given series, you need to first understand the pattern of the terms. You can then break down the series into separate series and use the formulas for the sum of an arithmetic series and the sum of a constant series to find the sum of each series. Finally, you can add the two sums together to find the sum of the original series.

References

[1] "Summation" by Wikipedia
[2] "Arithmetic Series" by Math Open Reference
[3] "Constant Series" by Math Is Fun

Introduction

In our previous article, we explored the concept of summation and how to find the sum of a given series. In this article, we will answer some frequently asked questions related to summation and series.

Q&A

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is given by: $S_n = \frac{n}{2}(a_1 + a_n)$

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

A: The formula for the sum of a constant series is given by: $S_n = na$

Q: How do I find the sum of a given series?

A: To find the sum of a given series, you need to first understand the pattern of the terms. You can then break down the series into separate series and use the formulas for the sum of an arithmetic series and the sum of a constant series to find the sum of each series. Finally, you can add the two sums together to find the sum of the original series.

Q: What is the difference between an arithmetic series and a constant series?

A: An arithmetic series is a series where each term is obtained by adding a fixed constant to the previous term. A constant series is a series where each term is a constant value.

Q: How do I determine if a series is an arithmetic series or a constant series?

A: To determine if a series is an arithmetic series or a constant series, you need to examine the pattern of the terms. If the terms are obtained by adding a fixed constant to the previous term, then it is an arithmetic series. If the terms are constant values, then it is a constant series.

Q: Can I use the formula for the sum of an arithmetic series to find the sum of a constant series?

A: No, you cannot use the formula for the sum of an arithmetic series to find the sum of a constant series. The formula for the sum of an arithmetic series is only applicable to arithmetic series, not constant series.

Q: Can I use the formula for the sum of a constant series to find the sum of an arithmetic series?

A: No, you cannot use the formula for the sum of a constant series to find the sum of an arithmetic series. The formula for the sum of a constant series is only applicable to constant series, not arithmetic series.

Q: How do I find the sum of a series with multiple terms?

A: To find the sum of a series with multiple terms, you need to break down the series into separate series and use the formulas for the sum of an arithmetic series and the sum of a constant series to find the sum of each series. Finally, you can add the two sums together to find the sum of the original series.

Q: Can I use a calculator to find the sum of a series?

A: Yes, you can use a calculator to find the sum of a series. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct formula.

Conclusion

In this article, we answered some frequently asked questions related to summation and series. We hope that this article has been helpful in clarifying any doubts you may have had about summation and series.

Frequently Asked Questions

What is the formula for the sum of an arithmetic series?
What is the formula for the sum of a constant series?
How do I find the sum of a given series?
What is the difference between an arithmetic series and a constant series?
How do I determine if a series is an arithmetic series or a constant series?
Can I use the formula for the sum of an arithmetic series to find the sum of a constant series?
Can I use the formula for the sum of a constant series to find the sum of an arithmetic series?
How do I find the sum of a series with multiple terms?
Can I use a calculator to find the sum of a series?

References

[1] "Summation" by Wikipedia
[2] "Arithmetic Series" by Math Open Reference
[3] "Constant Series" by Math Is Fun