The given arithmetic sequence is:
7, 5, 3, 1, -1...
To derive the recursive formula for this sequence, we can observe the following:
- The first term \( a_1 \) is 7.
- To find subsequent terms, we notice that each term decreases by 2 from the previous term.
The common difference \( d \) can be calculated as:
\[ d = a_2 - a_1 = 5 - 7 = -2 \]
In general, for an arithmetic sequence, the recursive formula can be expressed as:
\[ a_n = a_{n-1} + d \]
Using this information, we can define the recursive formula for the sequence as follows:
- Base case: \( a_1 = 7 \)
- Recursive case: \( a_n = a_{n-1} - 2 \) for \( n > 1 \)
So, the recursive formula for this arithmetic sequence is:
\[ \begin{align*} a_1 & = 7 \ a_n & = a_{n-1} - 2 \text{ for } n > 1 \end{align*} \]