What is the 8th term of the geometric sequence: -9, 36, -144, 576, -2304, 9216, -36864?

To determine the 8th term of the given geometric sequence, we first need to identify the common ratio between the consecutive terms. In a geometric sequence, each term is obtained by multiplying the previous term by a constant known as the common ratio.

Starting with the first two terms, we can find the common ratio by dividing the second term by the first term:

[

\text{Common Ratio} = \frac{36}{-9} = -4.

]

Next, we can verify this ratio by checking other consecutive terms. For example, from the second term to the third term:

[

\text{Common Ratio} = \frac{-144}{36} = -4,

]

and from the third term to the fourth term:

[

\text{Common Ratio} = \frac{576}{-144} = -4.

]

Since the common ratio remains consistent at -4, we can proceed to find the 8th term using the formula for the (n)th term of a geometric sequence:

[

a_n = a_1 \cdot r^{(n-1)},

]

where (a_1) is the first term, (r) is