Question

Find the 9th term of the geometric sequence 4,−16,64,...(1 point)
a9 =

Answer 1

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio (r).

To find the common ratio \( r \), we can take the second term and divide it by the first term:

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

We can also verify this by taking the third term and dividing it by the second term:

\[
r = \frac{64}{-16} = -4
\]

Now that we know the first term \( a_1 = 4 \) and the common ratio \( r = -4 \), we can find the \( n \)-th term of the geometric sequence using the formula:

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

We want to find the 9th term (\( n = 9 \)):

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

Now we calculate \((-4)^8\):

\[
(-4)^8 = 4^8 = 65536 \quad (\text{since } (-4)^{even} \text{ is positive})
\]

Now substitute back:

\[
a_9 = 4 \cdot 65536
\]

Calculate \( 4 \cdot 65536 \):

\[
a_9 = 262144
\]

Thus, the 9th term of the geometric sequence is:

\[
\boxed{262144}
\]