Question

The first six terms in a geometric sequence are shown, where "a" Subscript, 1 , Baseline equals negative 4.a1​=−4.

negative 4 , negative 16 , negative 64 , negative 256 , negative 1024 , negative 4096 ...−4, −16, −64, −256, −1024, −4096 ...

Based on this information, what is the ninth term in this sequence?

Answer 1

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:
\[
\boxed{-262144}
\]