91Ó°ÊÓ

In an arithmetic sequence, each term increases by a common difference from the previous one. The formula to find any term in the sequence, specifically the nth term, is given by: \[ a_n = a_1 + (n-1) \times d \] where:

• \( a_n \) is the nth term you're trying to find,
• \( a_1 \) is the first term of the sequence,
• \( n \) is the position of the term in the sequence, and
• \( d \) is the common difference between consecutive terms.

For instance, in the exercise where \( a_1 = 3b \), \( n = 7 \), and \( d = \frac{b}{3} \), you can find \( a_7 \) by substituting these values into the formula:\[ a_7 = 3b + (7-1) \times \frac{b}{3} \]This simplifies to:\[ a_7 = 3b + 2b = 5b \]Through this calculation, the 7th term of the sequence is determined. This formula is essential for identifying any term within an arithmetic sequence efficiently.

The sum of terms in an arithmetic sequence is a valuable calculation, especially when analyzing long series of numbers. The sum of the first \( n \) terms (\( S_n \)) is calculated using the formula:\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]This formula allows you to add up terms quickly by considering only the first and the nth term, rather than adding each term manually. Here:

• \( S_n \) stands for the sum of the sequence up to the nth term,
• \( n \) is the number of terms in the sequence,
• \( a_1 \) is the first term, and
• \( a_n \) is the nth term.

In the problem with \( a_1 = 3b \) and \( a_7 = 5b \), the sum of the first 7 terms is calculated as:\[ S_7 = \frac{7}{2} \times (3b + 5b) \]Combine and simplify to find:\[ S_7 = \frac{7}{2} \times 8b = 28b \]So, the sum of the first seven terms results in \( 28b \), showcasing the efficiency and practicality of using the sum formula.

Understanding the general term of a sequence is crucial for comprehending how arithmetic sequences work. It represents any term in the sequence based on its position. The general formula for finding a term in an arithmetic sequence is:\[ a_n = a_1 + (n-1) \times d \]It provides a straightforward way to compute any term without listing all previous ones.

• \( a_n \) represents the term you're finding,
• \( a_1 \) is the initial term,
• \( n \) indicates the position of the term, and
• \( d \) is the common difference.

For example, when dealing with the task of finding \( a_7 \) given \( a_1 = 3b \) and \( d = \frac{b}{3} \), the general term provides:\[ a_7 = 3b + 6 \times \frac{b}{3} \]This simplifies to:\[ a_7 = 5b \]The application of this formula provides insight into how each term is structured and is indispensable when dealing with larger sequences.