Arithmetic sequences are a specific type of sequence where the difference between consecutive terms remains constant. This constant difference is referred to as the common difference, denoted by the letter d. For example, in the sequence 2, 7, 12, 17, each term increases by 5, making the common difference d = 5.

To express an arithmetic sequence using a recursive formula, we can define the next term based on the previous term. The general form of the recursive formula is:

\( a_n = a_{n-1} + d \)

Here, \( a_n \) represents the current term, \( a_{n-1} \) is the previous term, and d is the common difference. For the sequence mentioned, the recursive formula would be:

\( a_n = a_{n-1} + 5 \)

To fully define the sequence, it is essential to specify the first term. In this case, the first term \( a_1 = 2 \).

For example, if we start with \( a_1 = 3 \) and have a common difference of 4, we can calculate the first four terms as follows:

• \( a_1 = 3 \)
• \( a_2 = a_1 + 4 = 3 + 4 = 7 \)
• \( a_3 = a_2 + 4 = 7 + 4 = 11 \)
• \( a_4 = a_3 + 4 = 11 + 4 = 15 \)

Thus, the first four terms are 3, 7, 11, and 15.

Conversely, if the common difference is negative, such as in the sequence starting with \( a_1 = 9 \) and a common difference of -6, the terms would be calculated as follows:

• \( a_1 = 9 \)
• \( a_2 = a_1 - 6 = 9 - 6 = 3 \)
• \( a_3 = a_2 - 6 = 3 - 6 = -3 \)
• \( a_4 = a_3 - 6 = -3 - 6 = -9 \)

In this case, the terms are 9, 3, -3, and -9.

When tasked with writing a recursive formula from a given sequence, the first step is to determine the common difference by subtracting any two consecutive terms. For instance, in the sequence 2, 5, 8, 11, the common difference is:

\( d = 5 - 2 = 3 \)

Thus, the recursive formula can be established as:

\( a_n = a_{n-1} + 3 \)

To complete the recursive formula, it is crucial to specify the first term, which in this case is \( a_1 = 2 \). Therefore, the complete recursive formula is:

\( a_n = a_{n-1} + 3, \quad a_1 = 2 \)

Understanding these concepts allows for the effective calculation and representation of arithmetic sequences through recursive formulas.