To complete the recursive formula of the arithmetic sequence given as ..., 3, -1, 1, 3, let's first identify the common difference and the initial term.
Looking at the sequence, we can calculate the common difference:
- From -1 to 1, the difference is 1 - (-1) = 2.
- From 1 to 3, the difference is 3 - 1 = 2.
The common difference \( d \) is 2.
Now, to identify the first term \( b(1) \): The sequence seems to start from 3, but since the sequence is provided in reverse order starting with 3, the first term when following the order shown would be 3.
So we have:
- \( b(1) = 3 \)
For the recursive part, the formula is:
- \( b(n) = b(n - 1) + d \)
Where \( d \) is the common difference which we found to be 2.
Putting it all together, we have:
- \( b(1) = 3 \)
- \( b(n) = b(n - 1) + 2 \) for \( n > 1 \)
Thus, the complete recursive formula for this arithmetic sequence is:
- \( b(1) = 3 \)
- \( b(n) = b(n - 1) + 2 \) for \( n > 1 \)