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An arithmetic series is a sum of terms in an arithmetic sequence. Each term in the sequence is formed by adding a constant difference to the previous term. The general form of an arithmetic sequence is: \ a, a + d, a + 2d, a + 3d, \ and so on, where \(a\) is the first term and \(d\) is the common difference.
In our exercise, the series \(2j\) forms an arithmetic sequence with the following terms: \[2 \cdot 0, 2 \cdot 1, 2 \cdot 2, 2 \cdot 3, 2 \cdot 4\]
This simplifies to: \[0, 2, 4, 6, 8\]
Here, \(d = 2\), which is the common difference between consecutive terms. This understanding helps in recognizing patterns and simplifying the summation process.

Summation notation, also called sigma notation, provides a shorthand way to represent the addition of a series of terms. It uses the Greek letter \(\sum\) (sigma) followed by an expression that tells us what to sum.
The general format is: \ \(\sum_{i=m}^{n}a_i\)
Here, \(i\) is the index of summation, \(m\) is the starting value, \(n\) is the ending value, and \(a_i\) is the expression for the terms to sum.
For example, in the given exercise: \ \(\sum_{j=0}^{4}(2j)\)
This means we start summing from \(j=0\) to \(j=4\), with each term being \(2j\). This notation helps in compactly expressing and manipulating series.

Understanding the steps to find the sum of a series is crucial in solving problems effectively.
Here are the key steps followed in our exercise:
Step 1 - Understand the Series:
Recognize the series and identify the expression for the terms. Identify the range of the index of summation.
Step 2 - List the Terms of the Series:
Substitute each value of the index (from the starting value to the ending value) into the expression to list all the terms. For example, substituting \(j\) from 0 to 4 into \(2j\) gives us terms: 0, 2, 4, 6, 8.
Step 3 - Calculate the Sum of the Terms:
Add all the listed terms together. In this case: \(0 + 2 + 4 + 6 + 8 = 20\).
By following these steps carefully, you can accurately find the sum of any arithmetic series presented in summation notation.