Why Options B and C are Incorrect

Oct 13, 2025 by windailys.com 48 views

Introduction

In mathematics, a series is a sequence of numbers that are added together to find a total sum. The sum of a series can be calculated using various methods, including arithmetic series, geometric series, and more. In this article, we will explore the sum of a given series and determine which of the provided options represents the correct sum.

The Given Series

The given series is:

${ 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 }$

This series consists of 11 terms, with each term increasing by 2.

Option A: Sum of the Series using Arithmetic Series Formula

Option A is represented by the formula:

${ S_{11} = \sum_{k=1}^{11}[6+(k-1) 2] }$

This formula uses the arithmetic series formula, which is:

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

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

Using this formula, we can calculate the sum of the series as follows:

${ S_{11} = \frac{11}{2}(6 + 26) }$

${ S_{11} = \frac{11}{2}(32) }$

${ S_{11} = 176 }$

Option B: Sum of the Series using Arithmetic Series Formula

Option B is represented by the formula:

${ S_{11} = \sum_{k=1}^{11}[2+(k-1) 6] }$

This formula also uses the arithmetic series formula, but with a different starting point.

Using this formula, we can calculate the sum of the series as follows:

${ S_{11} = \frac{11}{2}(2 + 26) }$

${ S_{11} = \frac{11}{2}(28) }$

${ S_{11} = 154 }$

Option C: Sum of the Series using Arithmetic Series Formula

Option C is represented by the formula:

${ S_{11} = \sum_{k=1}^{11}[4+(k-1) 8] }$

This formula also uses the arithmetic series formula, but with a different starting point.

Using this formula, we can calculate the sum of the series as follows:

${ S_{11} = \frac{11}{2}(4 + 36) }$

${ S_{11} = \frac{11}{2}(40) }$

${ S_{11} = 220 }$

Conclusion

Based on the calculations above, we can see that Option A represents the correct sum of the series:

${ S_{11} = 176 }$

This is because the arithmetic series formula used in Option A accurately calculates the sum of the series.

Why Option A is Correct

Option A is correct because it uses the arithmetic series formula, which is a well-established method for calculating the sum of a series. The formula takes into account the number of terms, the first term, and the last term, and provides an accurate calculation of the sum.

Why Options B and C are Incorrect

Options B and C are incorrect because they use the arithmetic series formula with different starting points, which results in an incorrect calculation of the sum.

Real-World Applications

The sum of a series has many real-world applications, including:

Finance: Calculating the total value of a portfolio or investment.
Engineering: Calculating the total weight or volume of a structure or material.
Science: Calculating the total energy or momentum of a system.

Conclusion

In conclusion, the sum of the given series is represented by Option A:

${ S_{11} = 176 }$

This is because the arithmetic series formula used in Option A accurately calculates the sum of the series. The sum of a series has many real-world applications, and understanding how to calculate it is essential for making accurate predictions and decisions.

References

Arithmetic Series Formula: A well-established method for calculating the sum of a series.
Mathematics: A branch of science that deals with numbers, quantities, and shapes.
Series: A sequence of numbers that are added together to find a total sum.

Q: What is a series in mathematics?

A: A series is a sequence of numbers that are added together to find a total sum. It can be a finite or infinite sequence of numbers.

Q: What is the arithmetic series formula?

A: The arithmetic series formula is a method for calculating the sum of a series. It is given by:

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

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

Q: How do I calculate the sum of a series using the arithmetic series formula?

A: To calculate the sum of a series using the arithmetic series formula, you need to follow these steps:

Identify the first term ( $a_1$ ) and the last term ( $a_n$ ) of the series.
Count the number of terms ( $n$ ) in the series.
Plug the values of $a_1$ , $a_n$ , and $n$ into the arithmetic series formula.
Simplify the expression to find the sum of the series.

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

A: An arithmetic series is a type of series that consists of terms that increase or decrease by a fixed amount. A geometric series is a type of series that consists of terms that increase or decrease by a fixed ratio.

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

A: To determine if a series is arithmetic or geometric, you need to examine the pattern of the terms. If the terms increase or decrease by a fixed amount, it is an arithmetic series. If the terms increase or decrease by a fixed ratio, it is a geometric series.

Q: What are some real-world applications of the sum of a series?

A: The sum of a series has many real-world applications, including:

Finance: Calculating the total value of a portfolio or investment.
Engineering: Calculating the total weight or volume of a structure or material.
Science: Calculating the total energy or momentum of a system.

Q: How do I use the sum of a series in finance?

A: To use the sum of a series in finance, you can calculate the total value of a portfolio or investment by summing up the individual values of the assets. For example, if you have a portfolio with 10 stocks, each with a value of $100, the total value of the portfolio would be:

${ S_{10} = 100 + 100 + 100 + ... + 100 }$

Using the arithmetic series formula, you can calculate the sum of the series as:

${ S_{10} = \frac{10}{2}(100 + 100) }$

${ S_{10} = 500 }$

Q: How do I use the sum of a series in engineering?

A: To use the sum of a series in engineering, you can calculate the total weight or volume of a structure or material by summing up the individual weights or volumes of the components. For example, if you have a building with 10 floors, each with a weight of 1000 kg, the total weight of the building would be:

${ S_{10} = 1000 + 1000 + 1000 + ... + 1000 }$

Using the arithmetic series formula, you can calculate the sum of the series as:

${ S_{10} = \frac{10}{2}(1000 + 1000) }$

${ S_{10} = 5000 }$

Q: How do I use the sum of a series in science?

A: To use the sum of a series in science, you can calculate the total energy or momentum of a system by summing up the individual energies or momenta of the components. For example, if you have a system with 10 particles, each with an energy of 100 J, the total energy of the system would be:

${ S_{10} = 100 + 100 + 100 + ... + 100 }$

Using the arithmetic series formula, you can calculate the sum of the series as:

${ S_{10} = \frac{10}{2}(100 + 100) }$

${ S_{10} = 500 }$

Conclusion

In conclusion, the sum of a series is a fundamental concept in mathematics that has many real-world applications. By understanding how to calculate the sum of a series, you can apply it to various fields, including finance, engineering, and science.