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When we talk about the _sum of a series_, especially in the context of an arithmetic series, we're discussing how to find the total of all the numbers in the sequence. An arithmetic series is a sequence of numbers that have a constant difference between consecutive terms. This sum tells us the total value we get when we add up all the terms together. For instance, if you have a series starting with 2 and ending at 200 with a total of 100 terms, you can quickly find its sum using a specific formula.

The formula to determine the sum of an arithmetic series is:

• \( S_n = \frac{n}{2} (a + l) \)

Where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( l \) is the last term, and \( n \) represents the number of terms. By applying the formula, you can efficiently get the sum without manually adding each number, which saves time and minimizes error. In the example of finding the sum of the first 100 positive even integers, applying this formula correctly gives us a total sum of 10,100.

The _common difference_ is a key feature of an arithmetic series. It is the difference between any two consecutive terms. For example, in the series 2, 4, 6, 8,... each term is obtained by adding a fixed number, which is the common difference, to the previous term. Here the common difference is 2.

To identify the common difference \( d \) in any series, you can subtract the first term from the second term. Ensuring uniformity across the series by checking subsequent terms is equally spaced is crucial. The common difference helps determine how far apart the terms are from each other and is essential for using other arithmetic formulas like finding the nth term or the sum of the series. Knowing \( d \) simplifies the understanding of the entire sequence's structure.

The _nth term formula_ is useful in identifying any term’s position within an arithmetic series, without having to list out each term. It allows you to find the exact value of the term at any position \( n \) in the series.

The formula for finding the nth term \( l \) of an arithmetic series is:

• \( l = a + (n - 1)d \)

Where \( l \) is the nth term, \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference. For instance, to find the 100th term in the series starting from 2 with a common difference of 2, simply plug in the values into the formula: \( l = 2 + (100 - 1) \times 2 = 200 \).

This formula ensures you can find any term quickly and is vital for calculating the sum of the series, as knowing the first and the last term are both necessary steps in that calculation.