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An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference. For example, in the sequence 2, 4, 6, 8, the common difference is 2.
In a general form, an arithmetic sequence can be defined as: \[a_n = a_1 + (n-1)d\] where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.
Summing an arithmetic series involves finding the sum of all terms from the first term to the nth term. The formula for the sum of an arithmetic series is: \[S_n = \frac{n}{2} (2a_1 + (n-1)d)\]
This equation simplifies the process of adding all the terms manually by making use of the patterns in arithmetic series.

Summation notation (also known as sigma notation) allows us to write long sums in a compact form. The symbol for summation is \(\sum\), and it is used to indicate that a sequence of numbers should be added together.
For instance, the notation \(\sum_{i=1}^{5} x_i\) means that we should sum the values of \(x_i\) from \(i=1\) to \(i=5\). It is essential to substitute each term into the formula before summing them up.
To evaluate the sum \(\sum_{i=1}^{5} \frac{x_i}{x_i+3}\), we substitute each value of \(x_i\):

• \(x_1 = -2 \rightarrow \frac{-2}{-2+3} = -2\)
• \(x_2 = -1 \rightarrow \frac{-1}{-1+3} = -0.5\)
• \(x_3 = 0 \rightarrow \frac{0}{0+3} = 0\)
• \(x_4 = 1 \rightarrow \frac{1}{1+3} = 0.25\)
• \(x_5 = 2 \rightarrow \frac{2}{2+3} = 0.4\)

After calculating these terms, you can easily sum them to evaluate the entire series: \(-2 + (-0.5) + 0 + 0.25 + 0.4 = -1.85\).

A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in a sequence is called a term. Sequences can be finite or infinite, depending on whether they have a limited number of terms or go on indefinitely.
There are different types of sequences in mathematics, including arithmetic sequences, geometric sequences, and more. In the given exercise, we deal with a finite arithmetic sequence.
Understanding sequences is crucial because many mathematical concepts and real-world applications depend on recognizing patterns and rules. When dealing with sequences, you often need to determine the general rule that defines the sequence, which can then be used for various operations, such as finding specific terms or summing the sequence.
In the given problem, the terms are provided: \(x_1 = -2\), \(x_2 = -1\), \(x_3 = 0\), \(x_4 = 1\), and \(x_5 = 2\). These terms follow an arithmetic pattern where each term increases by 1. Recognizing this pattern helps simplify calculations and understand the sequence's behavior.