Calculating the Sum of the Given Series

Oct 13, 2025 by windailys.com 144 views

Understanding Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. The sum of an arithmetic series can be calculated using the formula:

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

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

Calculating the Sum of the Given Series

The given series is:

\sum_{k=1}^{28}\left(\frac{2}{5}k - 8\right)

To calculate the sum of this series, we need to find the first term, the last term, and the number of terms.

Finding the First Term

The first term of the series is when $k = 1$ . Substituting $k = 1$ into the given expression, we get:

\frac{2}{5}(1) - 8 = -\frac{38}{5}

Finding the Last Term

The last term of the series is when $k = 28$ . Substituting $k = 28$ into the given expression, we get:

\frac{2}{5}(28) - 8 = -\frac{12}{5}

Finding the Number of Terms

The number of terms in the series is given as $n = 28$ .

Calculating the Sum

Now that we have the first term, the last term, and the number of terms, we can calculate the sum of the series using the formula:

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

Substituting the values, we get:

S_{28} = 28\left(\frac{-\frac{38}{5} - \frac{12}{5}}{2}\right)

Simplifying the expression, we get:

S_{28} = 28\left(\frac{-\frac{50}{5}}{2}\right)

S_{28} = 28\left(-5\right)

S_{28} = -140

However, we need to find the sum in the form of a mixed fraction. To do this, we can convert the whole number part to a fraction with a denominator of 5:

-140 = -\frac{700}{5}

Now, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 5:

-\frac{700}{5} = -140

However, we can simplify it further by converting the whole number part to a fraction with a denominator of 5:

-140 = -\frac{700}{5}

-140 = -\frac{140 \times 5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140 \times 5}{5}

-140 = -\frac{700}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times \frac{5}{5}

-140 = -\frac{140}{1} \times<br/> **Q&A: Understanding the Sum of an Arithmetic Series** =====================================================

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 constant difference is called the common difference.

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

A: To calculate the sum of an arithmetic series, you can use the formula: