Question

the first six terms in a geometric sequence are shown, where a1 = -4
-4, -16, -64, -256, -1024, -4096...
based on this information, what is the ninth term in this sequence?
9th term: ________

Answer 1

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed number called the common ratio.

Given the first term \( a_1 = -4 \), we can identify the common ratio by dividing the second term by the first term:

\[
r = \frac{a_2}{a_1} = \frac{-16}{-4} = 4
\]

Now that we have the common ratio \( r = 4 \), we can find the formula for the \( n \)-th term of the geometric sequence, which is given by:

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

To find the ninth term \( a_9 \):

\[
a_9 = a_1 \cdot r^{(9-1)} = -4 \cdot 4^{8}
\]

Calculating \( 4^8 \):

\[
4^8 = (4^2)^4 = 16^4
\]

Calculating \( 16^2 = 256 \):

\[
16^4 = 256^2 = 65536
\]

Now, substitute that back into the equation for \( a_9 \):

\[
a_9 = -4 \cdot 65536 = -262144
\]

Thus, the ninth term in the geometric sequence is

\[
\boxed{-262144}
\]