91Ó°ÊÓ

In arithmetic sequences, the first term is fundamental. It is the starting point for the entire sequence. This term is represented by \(a_{1}\).

• In the given problem, our first term is 43.
• This indicates that the sequence begins with the number 43.

Beginning a sequence with the first term means each subsequent term is built upon it, using the common difference. Without knowing the first term, constructing the sequence accurately is impossible. The first term sets the stage for what numbers will follow in the progression of the sequence.

The common difference, denoted as \(d\), is what sets arithmetic sequences apart. It is a constant difference between consecutive terms.

• In our example, the common difference is \(-7\).
• This means each term is 7 less than the term before it.

This negative common difference indicates that the sequence is decreasing. The same value is subtracted repeatedly to find the next term. Understanding the common difference helps in grasping the sequence's pattern and direction.

A sequence formula in arithmetic sequences provides a comprehensive rule to find any term in the sequence. The general formula is:\[ a_{n} = a_{1} + (n-1) \times d \]Here: - \(a_{n}\) is the term we want to find. - \(a_{1}\) is the first term. - \(d\) is the common difference. - \(n\) is the number of the term we seek.Using this formula, we can determine any term. We start from the first term and include the effect of the common difference over \(n-1\) increments.In our original exercise, substituting the given values into this formula, we get:\[ a_{n} = 43 + (n-1) \times (-7) \]This formula allows us to calculate any term in the sequence, such as the 5th term or the 10th term, with ease. The arithmetic sequence formula helps students to see the connections between terms clearly and predict future terms.