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The explicit formula is a powerful tool in arithmetic sequences because it allows you to find any term directly without having to calculate all the preceding terms. In an arithmetic sequence, each term after the first is generated by adding the same fixed number, known as the common difference.

The general explicit formula for an arithmetic sequence is given by:

• \( a_n = a_1 + (n-1)d \)

Here, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. For instance, with the sequence \(-2, 5, 12, 19, 26, \ldots\), the first term \( a_1 \) is \(-2\) and the common difference \( d \) is \(7\). If we plug these values into the formula, we get:

• \( a_n = -2 + (n-1)7 \)

Simplifying gives:

• \( a_n = 7n - 9 \)

This formula means you can easily find the 10th term, the 50th term, or any term by simply plugging in the value of \( n \). It's efficient and crucial for working with large sequences.

A recursive formula provides a way to determine the next term in a sequence using the previous term. It is particularly helpful when you are only interested in a few terms and want to establish a relationship between consecutive terms.

The recursive formula generally takes the form:

• \( a_n = a_{n-1} + d \)

Here, \( a_{n-1} \) is the previous term, and \( d \) is the common difference. For our sequence, where \(-2\) is the first term and the common difference is \(7\), the recursive formula becomes:

• \( a_n = a_{n-1} + 7 \)

Let's say you know the first term is \(-2\), then:

• The second term \( a_2 = a_1 + 7 = -2 + 7 = 5 \)
• The third term \( a_3 = a_2 + 7 = 5 + 7 = 12 \)

And so forth. This showcases how each term builds upon the last, offering a step-by-step approach for sequences when you need to track progression.

In an arithmetic sequence, the common difference is the consistent interval between each successive term. It's denoted by \(d\) and it plays a key role in both explicit and recursive formulas.

Subtract any term from the next term to find it. For example, in the sequence \(-2, 5, 12, 19, 26, \ldots\), subtract the first term from the second:

• \( 5 - (-2) = 7 \)

This indicates a common difference of \(7\). To ensure accuracy, check several pairs of consecutive terms:

• \( 12 - 5 = 7 \)
• \( 19 - 12 = 7 \)
• \( 26 - 19 = 7 \)

The consistent result signifies that the common difference remains 7 throughout, confirming our sequence is indeed arithmetic. Knowing the common difference is essential for constructing formulas and understanding how the sequence evolves.