91Ó°ÊÓ

In arithmetic sequences, the explicit rule is an essential tool used to find the value of any term in a sequence without having to list all preceding terms. This rule allows us to express the general term, denoted as \(a_n\), directly in terms of its position \(n\). The explicit rule for an arithmetic sequence can be derived from the formula:

• \(a_n = a_1 + (n-1) \cdot d\)

where:

• \(a_n\) is the nth term of the sequence.
• \(a_1\) is the first term of the sequence.
• \(n\) is the term number.
• \(d\) is the common difference between consecutive terms.

In our exercise, the explicit rule was derived using the first term \(a_1 = -12\) and the common difference \(d = 9.1\). By substituting these values into the formula, we obtain the explicit rule: \[a_n = -12 + (n-1) \cdot 9.1\] This rule allows us to easily compute the value of any term in the sequence by simply plugging in the desired position \(n\) into the equation.

The common difference is a key characteristic in identifying and working with arithmetic sequences. It represents the constant amount added to each term to generate the next term in the sequence. This concept is crucial because if the difference between consecutive terms is not constant, the sequence is not arithmetic.The common difference \(d\) can be determined by subtracting any term in the sequence from the subsequent term. For our sequence, this is given by:

• \(d = a_{n} - a_{n-1}\)

In the exercise, the common difference was indicated as 9.1. This means that each subsequent term increases by 9.1 compared to the previous one. Knowing the common difference helps in predicting future terms and aids in forming the explicit rule, which directly incorporates this constant change between terms.

Determining the sequence type is a crucial first step in solving any sequence-related problem. The type of sequence dictates the approach for finding elements and predicting future terms. In our exercise, we identified the sequence as arithmetic. Here's why:

• An arithmetic sequence is defined by a constant difference between consecutive terms.
• If we compare each term to the next, the difference should remain the same throughout.

In the sequence presented in the problem, each term increases by a fixed amount, 9.1, confirming its classification as an arithmetic sequence. Recognizing the sequence type early on simplifies the process of writing an explicit rule, as each sequence type has its unique formula for deriving terms based on their position.