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An arithmetic sequence is a list of numbers with a definite pattern. Each term after the first is calculated by adding a consistent number called the common difference to the previous term.

For example, in the given sequence \( -6, -1, 4, 9, \dots \), the common difference is found by subtracting the first term from the second term. We can express this as \( d = -1 - (-6) = 5 \). This difference is constant throughout the sequence, meaning every subsequent term increases by five. So, the sequence progresses as \( -1 + 5 = 4 \), then \( 4 + 5 = 9 \), and so on.

Understanding the components of an arithmetic sequence allows us to anticipate future terms and provides the foundation to tackle more complex problems, such as finding the sum of a specific number of terms within the sequence.

The sum of an arithmetic series is the total of all terms in the sequence. To avoid the tedium of adding each term individually, especially in long sequences, mathematicians have derived a formula to directly calculate the sum.

The problem prompts us to find the sum of the first 14 terms of the series. To do this effectively, we don't need to calculate each term; instead, we use the formula \( S_n = \frac{n}{2} \times (2a + (n-1)d) \) where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms to be added.

In our example, for the sequence \( -6, -1, 4, 9, \dots \) the sum of the first 14 terms is calculated by recognizing that the first term \( a \) is \( -6 \) and the common difference \( d \) is \( 5 \) as identified in the solution steps. By substituting these values into the formula, we can find the desired sum without manually adding each term.

The arithmetic series formula is a key concept in algebra that allows quick calculation of the sum of the terms in an arithmetic sequence. The general form of the formula, \( S_n = \frac{n}{2} \times (2a + (n-1)d) \), is used to find the sum \( S_n \) when we know the number of terms to be added \( n \) and the sequence's first term \( a \) and common difference \( d \).

To illustrate, let's dissect the given solution where the first term, \( a \) is \( -6 \) and the common difference, \( d \) is \( 5 \) and we want to find the sum of the first 14 terms \( S_{14} \). By substituting the known values into the formula, we get \( S_{14} = \frac{14}{2} \times (2 \times (-6) + (14 - 1) \times 5) = 7 \times (28) = 196 \). It is important to follow the order of operations carefully during substitution to ensure accuracy.

By mastering the arithmetic series formula, students can save time and reduce potential errors in calculation, making it an indispensable tool for solving problems involving arithmetic sequences.