91Ó°ÊÓ

The common difference, denoted by the letter \(d\), is a fundamental characteristic of an arithmetic sequence. It reflects the consistent interval between consecutive terms. To find the common difference in a sequence, you subtract any term from the term that follows it. In our exercise, we know two specific terms: \(a_{12} = 42\) and \(a_{8} = 26\). To calculate \(d\), subtract \(a_{8}\) from \(a_{12}\), then divide by the difference in their term indices, i.e., 12 minus 8:\[d = \frac{42 - 26}{12 - 8} = 4\]

• Give any consecutive pair of terms in an arithmetic sequence, and the difference between their indices.
• This difference will always be the same throughout the arithmetic sequence.

The arithmetic sequence formula provides a method to find any term in the sequence if you know the first term and the common difference. The general formula is:\[a_n = a_1 + (n - 1) \cdot d\]Here, \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. In our exercise, we needed to use known terms, specifically \(a_8 = 26\), to work backwards and determine \(a_1\). This is completed by rearranging the formula:\[a_1 = a_8 - (8 - 1) \cdot d = 26 - 7 \cdot 4 = -2\]Using the arithmetic sequence formula allows you to find any specific term without needing to list all prior terms. It's particularly useful for finding terms far along in the sequence.

The first term of an arithmetic sequence, symbolized as \(a_1\), is the starting point of the sequence. Knowing this term helps in determining the entire sequence. In exercises dealing with arithmetic sequences, sometimes you are provided mid-sequence terms. In our problem, after finding the common difference, we calculated \(a_1\) using the term \(a_8\) and the common difference:\[a_1 = a_8 - (8 - 1) \cdot d = 26 - 28 = -2\]Once \(a_1\) is known, finding subsequent terms is straightforward. For our exercise, the sequence begins with the term \(-2\).

• The first term is crucial as it sets the sequence in motion.
• Without it, you cannot establish subsequent terms accurately using the formula \(a_n = a_1 + (n - 1) \cdot d\).