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When working with arithmetic sequences, a common task is calculating the sum of its terms. The sum of an arithmetic sequence can be found using a convenient formula. This formula allows you to efficiently find the sum without having to add each term individually.

An arithmetic sequence is characterized by a constant difference between consecutive terms. In our example, the sequence is 4, 8, 12, 16, 20, and so on. To find the sum of the first 14 terms, which is what we aim to do here, we can use the sum formula:

• The formula for the sum of the first \(n\) terms is: \(S_n = \frac{n}{2}(2a + (n-1)d)\)
• \(a\) is the first term.
• \(d\) represents the common difference.

This formula derives from the idea that you can pair the sequence terms from the start and the end to minimize manual calculations. By finding this sum, repetitive addition is greatly simplified, making it a highly efficient approach.

The common difference in an arithmetic sequence is the amount that each term increases (or decreases) from the previous term. It is a crucial part of understanding the structure of arithmetic sequences.

In our sequence example, the numbers are 4, 8, 12, 16, and 20. Each of these numbers is separated by the same difference, which is 4. You can find the common difference by subtracting any term from the term that follows it. For instance:

• \(8 - 4 = 4\)
• \(12 - 8 = 4\)

This constant difference ensures our sequence is arithmetic. Knowing this pattern is vital because it allows the predictability needed for using the sequence formulas. It's what makes those formulas work!

The first term of an arithmetic sequence is often denoted by \(a\). It is the starting point of the sequence and serves as a baseline for establishing the pattern defined by the common difference.

In our case, the sequence starts with 4, which means the first term \(a = 4\). This term is important because it helps in determining other terms and the sum of the sequence:

• It sets the initial value from which the sequence grows.
• Used in the formula: \(S_n = \frac{n}{2}(2a + (n-1)d)\).

Understanding the first term helps you grasp the sequence structure more clearly, and it is essential for any calculations related to that sequence.

The \(n\)-th term of an arithmetic sequence gives the value of a specific term when you know its position \(n\). This requires the first term \(a\) and the common difference \(d\).

The formula for the \(n\)-th term is:

• \(l = a + (n-1)d\)
• Here, \(l\) refers to the \(n\)-th term.

For example, in our arithmetic sequence, you need to find the 14th term or any other specific term. With \(a = 4\), \(d = 4\), and \(n = 14\):

The calculation becomes:

• \(l = 4 + (14-1)\times4 = 4 + 52 = 56\)

This formula helps determine any term in the sequence, making it a powerful tool in sequence analysis. It ensures step-by-step predictability within the arithmetic framework.