To find the recursive formula for the given geometric sequence, we first need to determine the common ratio \( r \).
The sequence given is: -19, -13, -1, -3, -9, ...
We can find the common ratio \( r \) by dividing any term by its preceding term.
Let's calculate the ratio for the first two terms: \[ r = \frac{a_2}{a_1} = \frac{-13}{-19} = \frac{13}{19} \]
Now, let's check if this ratio holds for the following terms: \[ r = \frac{a_3}{a_2} = \frac{-1}{-13} = \frac{1}{13} \] This does not match with the first ratio, indicating that this is not a geometric sequence.
In fact, the given sequence appears to be an arithmetic sequence rather than a geometric sequence. A geometric sequence has a common ratio, while an arithmetic sequence has a common difference.
Next, we will look for the common difference \( d \): \[ d = a_2 - a_1 = -13 - (-19) = -13 + 19 = 6 \] \[ d = a_3 - a_2 = -1 - (-13) = -1 + 13 = 12 \] \[ d = a_4 - a_3 = -3 - (-1) = -3 + 1 = -2 \]
Clearly, we see that the difference is not consistent.
To conclude, the original options are incorrect since the sequence does not appear to be geometric, and no common ratio can be found.
If you provide more clear instructions or verify the sequence terms, I can assist you further. However, based on what's given, the correct recursive formula is not present in the original options provided.