To find the recursive formula for the arithmetic sequence given (7, 5, 3, 1, -1...), we first identify the common difference.
The sequence is:
- The first term \( a_1 = 7 \)
- The second term \( a_2 = 5 \)
- The third term \( a_3 = 3 \)
- The fourth term \( a_4 = 1 \)
- The fifth term \( a_5 = -1 \)
To find the common difference \( d \), we subtract consecutive terms: \( a_2 - a_1 = 5 - 7 = -2 \)
Checking other pairs: \( a_3 - a_2 = 3 - 5 = -2 \) \( a_4 - a_3 = 1 - 3 = -2 \) \( a_5 - a_4 = -1 - 1 = -2 \)
The common difference \( d \) is \( -2 \).
Now we can write the recursive formula for the sequence. The recursive definition expresses each term based on the previous term:
\[ \begin{align*} a_1 &= 7 \ a_n &= a_{n-1} - 2 \quad \text{for } n \geq 2 \end{align*} \]
Thus, the recursive formula for the given arithmetic sequence is:
\[ \begin{align*} a_1 &= 7 \ a_n &= a_{n-1} - 2 \quad \text{for } n \geq 2 \end{align*} \]