An Arithmetic Series Is Represented By The Equation S 24 = ∑ K = 1 24 ( − 6 + 0.5 K S_{24}=\sum_{k=1}^{24}(-6+0.5k S 24 ​ = ∑ K = 1 24 ​ ( − 6 + 0.5 K ]. Which Of The Following Is True?A. The Value Of The Series Is Greater Than The Value Of The 24th Number In The Series.B. The Value Of The Series Is Equal To

Introduction to Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. This type of series is commonly used in mathematics, particularly in the study of sequences and series. In this article, we will explore the equation $S_{24}=\sum_{k=1}^{24}(-6+0.5k)$ and determine which of the given statements is true.

Understanding the Equation

The given equation represents an arithmetic series with 24 terms. The general form of an arithmetic series is $S_n = \sum_{k=1}^{n} (a + (k-1)d)$, where $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms. In this case, the equation can be rewritten as $S_{24} = \sum_{k=1}^{24} (-6 + 0.5k)$.

Calculating the Sum of the Series

To calculate the sum of the series, we can use the formula for the sum of an arithmetic series: $S_n = \frac{n}{2}(2a + (n-1)d)$. In this case, $a = -6$, $d = 0.5$, and $n = 24$. Plugging these values into the formula, we get:

$S_{24} = \frac{24}{2}(2(-6) + (24-1)0.5)$

$S_{24} = 12(-12 + 11.5)$

$S_{24} = 12(-0.5)$

$S_{24} = -6$

Analyzing the Statements

Now that we have calculated the sum of the series, let's analyze the given statements:

A. The value of the series is greater than the value of the 24th number in the series.

B. The value of the series is equal to the value of the 24th number in the series.

Determining the Correct Statement

To determine which statement is true, we need to find the value of the 24th number in the series. We can do this by plugging $k = 24$ into the equation:

$(-6 + 0.5(24))$

$(-6 + 12)$

$6$

Since the value of the series is $-6$ and the value of the 24th number is $6$, statement A is true.

Conclusion

In conclusion, the value of the series is greater than the value of the 24th number in the series. This is because the sum of the series is $-6$, while the value of the 24th number is $6$. We hope this article has provided a clear understanding of the equation and its properties.

Frequently Asked Questions

Q: What is an arithmetic series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant.

Q: How do you calculate the sum of an arithmetic series?

A: You can use the formula $S_n = \frac{n}{2}(2a + (n-1)d)$, where $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.

Q: What is the value of the 24th number in the series?

A: The value of the 24th number in the series is $6$.

Q: Which statement is true?

A: Statement A is true, which states that the value of the series is greater than the value of the 24th number in the series.

References

• [1] "Arithmetic Series." MathWorld, Wolfram Research.
• [2] "Sequences and Series." Mathematics, 2nd ed., McGraw-Hill Education.

Further Reading

• "Arithmetic Series: A Comprehensive Guide"
• "Sequences and Series: A Mathematical Perspective"

Note: The references and further reading section are for additional information and resources on the topic. They are not directly related to the article's content.

Introduction

Arithmetic series are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and calculus. In this article, we will address some of the most frequently asked questions about arithmetic series, providing clear and concise answers to help you better grasp this concept.

Q&A

Q: What is an arithmetic series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. This type of series is commonly used in mathematics, particularly in the study of sequences and series.

Q: How do you calculate the sum of an arithmetic series?

A: You can use the formula $S_n = \frac{n}{2}(2a + (n-1)d)$, where $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.

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

A: The formula for the nth term of an arithmetic series is $a_n = a + (n-1)d$, where $a$ is the first term, $d$ is the common difference, and $n$ is the term number.

Q: How do you find the common difference of an arithmetic series?

A: To find the common difference of an arithmetic series, you can use the formula $d = \frac{a_n - a_{n-1}}{2}$, where $a_n$ is the nth term and $a_{n-1}$ is the (n-1)th term.

Q: What is the formula for the sum of the first n natural numbers?

A: The formula for the sum of the first n natural numbers is $S_n = \frac{n(n+1)}{2}$.

Q: How do you find the sum of an arithmetic series with a negative common difference?

A: To find the sum of an arithmetic series with a negative common difference, you can use the formula $S_n = \frac{n}{2}(2a + (n-1)d)$, where $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.

Q: What is the relationship between the sum of an arithmetic series and the average of its terms?

A: The sum of an arithmetic series is equal to the average of its terms multiplied by the number of terms. This can be expressed as $S_n = \frac{n}{2}(a + l)$, where $a$ is the first term, $l$ is the last term, and $n$ is the number of terms.

Q: How do you find the average of an arithmetic series?

A: To find the average of an arithmetic series, you can use the formula $\frac{S_n}{n}$, where $S_n$ is the sum of the series and $n$ is the number of terms.

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

A: The formula for the sum of an infinite arithmetic series is $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio.

Q: How do you determine if an arithmetic series is convergent or divergent?

A: An arithmetic series is convergent if the absolute value of the common ratio is less than 1, and divergent otherwise.

Conclusion

In conclusion, arithmetic series are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and calculus. We hope this article has provided a clear and concise overview of the most frequently asked questions about arithmetic series.

Frequently Asked Questions (FAQs)

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

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant, while a geometric series is a sequence of numbers in which the ratio between any two consecutive terms is constant.

Q: How do you use arithmetic series in real-life applications?

A: Arithmetic series are used in various real-life applications, such as finance, economics, and engineering.

Q: What are some common mistakes to avoid when working with arithmetic series?

A: Some common mistakes to avoid when working with arithmetic series include:

• Not checking for convergence or divergence
• Not using the correct formula for the sum of an arithmetic series
• Not considering the common difference or ratio

References

• [1] "Arithmetic Series." MathWorld, Wolfram Research.
• [2] "Sequences and Series." Mathematics, 2nd ed., McGraw-Hill Education.

Further Reading

• "Arithmetic Series: A Comprehensive Guide"
• "Sequences and Series: A Mathematical Perspective"

Note: The references and further reading section are for additional information and resources on the topic. They are not directly related to the article's content.