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An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This constant is known as the common difference.
In our given example, the sequence is: 3, 6, 9, 12, 15, 18. Here, the common difference is 3. You find the common difference by subtracting any term from the term that follows it. For instance, 6 - 3 = 3 or 9 - 6 = 3.
Understanding the common difference allows us to know how the sequence progresses and to predict future terms in the sequence.

**Key Points:**

• Arithmetic sequences follow a simple pattern of adding a fixed number to each term.
• Identifying the first term and the common difference is crucial in working with arithmetic sequences.

The general term in an arithmetic sequence allows us to find any term in the sequence without listing all previous terms. The formula for the general term (also known as the nth term) of an arithmetic sequence is:
\(a_n = a_1 + (n - 1) \times d\), where

• \(a_n\) is the nth term,
• \(a_1\) is the first term in the sequence,
• \(n\) is the position of the term in the sequence,
• \(d\) is the common difference.

Using our example: 3, 6, 9, 12, 15, 18, we identified the first term \(a_1\) as 3 and the common difference \(d\) as 3.
Thus, the general term \(a_n\) is determined as:
\[a_n = 3 + (n-1) \times 3 = 3n\]
This formula allows you to find any term in the sequence by simply plugging in the value of \(n\). For example, if you want to find the 4th term, substitute \(n = 4\) into the formula to get \[a_4 = 3 \times 4 = 12\].

**Key Points:**

• The general term formula makes it easy to find any term in the sequence without the need to list out all terms.
• Understanding the general term is essential for working with arithmetic sequences.

Summation notation is a way to represent the sum of a series of terms using a specific notation called sigma (\( \Sigma \) notation). It helps simplify the representation of the sum of large sequences.
In our example, the series 3, 6, 9, 12, 15, 18 has 6 terms. Using the general term \(a_n = 3n\), we can use summation notation to write the sum of these terms as:
\[ \sum_{n=1}^{6} 3n \]
This means you add up the values of \(3n\) from \(n = 1\) to \(n = 6\): 3(1) + 3(2) + 3(3) + 3(4) + 3(5) + 3(6).

**Key Points:**

• Summation notation provides a compact and clear way to express the addition of many terms.


• Understanding summation notation is crucial for working efficiently with series and sequences.