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Arithmetic sequences are everywhere. They are patterns of numbers that add a constant value, known as the common difference, each time. Knowing how to find the sum of such sequences is vital.
When we want to find the sum of the first n terms of an arithmetic sequence, we use a specific formula:

\( S_n = \frac{n}{2} \cdot (2a_1 + (n - 1)d) \)

Here, \(S_n\) is the sum of the first n terms, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.
Understanding this formula helps you easily calculate the sum without having to add each term one by one. Let's break it down.

• First, find the number of terms \(n\).
• Then, determine the first term \(a_1\) and the common difference \(d\).
• Plug the values into the formula to get the sum \(S_n\).

For our given exercise with \(a_1 = 6\), \(d = 3\), and \(n = 6\), it works like this:

\(S_6 = \frac{6}{2} \cdot (2 \cdot 6 + (6 - 1) \cdot 3)\)
This simplifies to:
\( S_6 = 3 \cdot (12 + 15) = 3 \cdot 27 = 81 \)
Therefore, the sum \(S_6\) is 81.

The common difference in an arithmetic sequence is the value that you add to each term to get the next term.
The formula to find the common difference is:\( d = a_{n+1} - a_n \)
This difference stays the same through the entire sequence. In our exercise, the common difference is 3. Let's see how it works:

• If the first term \(a_1\) is 6, then the second term \(a_2\) is \(6 + 3 = 9\).
• The third term \(a_3\) is \(9 + 3 = 12\), and so on.
• This pattern continues, adding 3 to each term to get the next term.

The common difference is crucial because it allows us to know how the sequence grows. By understanding and identifying the common difference, we can easily predict any term in the sequence.

The first term of an arithmetic sequence is vitally important. It sets the starting point for the entire sequence. We denote the first term by \(a_1\). In our problem, \(a_1\) is given as 6.
Here's why the first term matters:

• It determines the initial value in your arithmetic pattern.
• All subsequent terms are calculated from this point using the common difference \(d\).
• Changing \(a_1\) changes the entire sequence, making it crucial for accurate calculations.

For instance, if \(a_1\) was 10 instead of 6, the sequence would start at 10 and follow the common difference from there. In our exercise, starting with 6 and a common difference of 3, the first few terms would be 6, 9, 12, and so forth.
Always start with the first term \(a_1\) in mind when dealing with arithmetic sequences.