To find the ninth term of the given geometric sequence, we first need to identify the common ratio.
The first term \( a_1 \) is given as: \[ a_1 = -4 \]
The second term \( a_2 \) is: \[ a_2 = -16 \]
To find the common ratio \( r \), we use the formula for the common ratio in a geometric sequence: \[ r = \frac{a_2}{a_1} = \frac{-16}{-4} = 4 \]
Now that we have the common ratio \( r = 4 \), we can express the nth term of a geometric sequence as: \[ a_n = a_1 \cdot r^{n-1} \]
To find the ninth term \( a_9 \), we substitute \( n = 9 \): \[ a_9 = a_1 \cdot r^{9-1} = -4 \cdot 4^{8} \]
Next, we calculate \( 4^8 \): \[ 4^8 = (4^4)^2 = 256^2 \] Calculating \( 256^2 \): \[ 256^2 = 65536 \]
Therefore, we have: \[ a_9 = -4 \cdot 65536 = -262144 \]
Thus, the ninth term in the sequence is: \[ \boxed{-262144} \]