Question

The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary).
32, comma, 16, comma, 8, comma, point, point, .
32,16,8,...
Find the 9th term.
Find the 9th term.

Answer 1

To determine the 9th term of the sequence \( 32, 16, 8, \ldots \), we first need to identify the pattern of the sequence.

The sequence appears to be a geometric sequence where each term is half of the previous term. We can express this mathematically:

- The first term \( a_1 = 32 \)
- The second term \( a_2 = 16 \) (which is \( 32 \times \frac{1}{2} \))
- The third term \( a_3 = 8 \) (which is \( 16 \times \frac{1}{2} \))

We can see that each term can be described by the formula for the \( n \)-th term of a geometric sequence:

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

where \( a_1 \) is the first term and \( r \) is the common ratio. In this case, \( r = \frac{1}{2} \).

Now, we want to find the 9th term \( a_9 \):

\[
a_9 = 32 \times \left(\frac{1}{2}\right)^{(9-1)} = 32 \times \left(\frac{1}{2}\right)^8
\]

Calculating \( \left(\frac{1}{2}\right)^8 \):

\[
\left(\frac{1}{2}\right)^8 = \frac{1}{2^8} = \frac{1}{256}
\]

Therefore, we find \( a_9 \):

\[
a_9 = 32 \times \frac{1}{256} = \frac{32}{256} = \frac{1}{8} = 0.125
\]

Thus, the 9th term of the sequence is:

\[
\boxed{0.125}
\]