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When you first look at an arithmetic sequence, you will notice that it is a list of numbers with a consistent pattern. This pattern is defined by the **common difference**. In each arithmetic sequence, the common difference is the amount you add (or subtract) to get from one term to the next. It must remain constant throughout the sequence.

In our example sequence, the numbers are 3, 7, 11, 15, and 19. To find the common difference, subtract the first term from the second. Here, you perform the calculation: \(d = 7 - 3 = 4\). This means, for this sequence, every term is obtained by adding 4 to the previous one.

Identifying the common difference is crucial because it helps us understand the growth of the sequence.

• For a sequence that increases, the common difference will be positive.
• For a sequence that decreases, it will be negative.
• A common difference of zero means all terms are the same.

An explicit formula is like a rule that allows you to calculate the nth term of a sequence without listing all the previous terms. For any arithmetic sequence, the explicit formula follows this structure: \(a_n = a_1 + (n-1) \times d\).

Here:

• \(a_n\) represents the nth term.
• \(a_1\) is the first term of the sequence.
• \(n\) is the term number you're interested in.
• \(d\) is, as always, the common difference.

Using the example sequence where \(a_1 = 3\) and \(d = 4\), the explicit formula becomes \(a_n = 3 + (n-1) \times 4\).

This formula is immensely helpful because it offers a direct path to find any term in the sequence without needing to calculate all the preceding terms.

The nth term in a sequence gives us the value of any term at the position we specify as "n". By using the explicit formula, we can directly calculate this value, which saves time and effort, especially when dealing with large or infinite sequences.

For our earlier example, we need to find the 12th term, or \(a_{12}\). By substituting 12 for \(n\) in the formula \(a_n = 3 + (n-1) \times 4\), you can calculate:\[a_{12} = 3 + (12 - 1) \times 4 = 3 + 11 \times 4 = 47\]

This shows that the 12th term of the sequence is 47.

• The explicit formula allows us to bypass calculating each step directly.
• It's efficient for sequences with straightforward patterns like arithmetic sequences.